Q:

What is the evidence for the Trump/Russia investigation?

I'm pretty confused about the whole thing. All I've really heard about it is people's insistence that the Trump campaign colluded with the Russian government to sway the 2016 presidential election. I'm not even sure if the FBI was officially investigating the issue or not. So what started the whole thing? There must be some little tidbit of evidence somewhere that convinced everyone to start talking about it. What is that little bit of evidence that started this whole thing? Then from there, what other evidence has surfaced?

A:

I'm not even sure if the FBI was officially investigating the issue or not.

Yes, the FBI was officially (and likely is) investigating the issue. James Comey confirmed that during a hearing before the House Intelligence Committee. The investigation is still on at the time he was dismissed from the FBI.

Mr. Comey said the F.B.I. was “investigating the nature of any links between individuals associated with the Trump campaign and the Russian government, and whether there was any coordination between the campaign and Russia’s efforts.”
Source: https://www.nytimes.com/2017/03/20/us/politics/fbi-investigation-trump-russia-comey.html

So what started the whole thing?

There are various speculations that the UK's intelligence agency GCHQ alerted its US counterparts to possible contacts.
The DNC emails hack may also have prompted the investigation into whether the Russians interfered with the US election. The US intelligence agencies seemed to have traced them back to Russia.
Officials of the Trump campaign had also purportedly met with Russian officials during the campaign. Though this itself is not illegal, the investigation aims to find out the potential ties and links between them. Trump's campaign manager Paul Manafort, who resigned eventually, was also accused of having close ties to Russia. Also, Sessions didn't reveal during his confirmation hearing of his purported meetings between him and Russian officials. In response, he recused himself from the FBI investigation into the Russian hacking claims which he is overseeing.
Regarding evidence, there's no publicly available evidence that Trump colluded with the Russian government. However, the Office of the Director of National Intelligence (ODNI) released a declassified report in January 2017 regarding its findings that Russia tried to interfere with the election.

We assess Russian President Vladimir Putin ordered an influence campaign in 2016 aimed at the US presidential election. Russia’s goals were to undermine public faith in the US democratic process, denigrate Secretary Clinton, and harm her electability and potential presidency. We further assess Putin and the Russian Government developed a clear preference for President-elect Trump. We have high confidence in these judgments.

A:

Here's what we know
Several prominent Democrats and their assistants (most notably James Podesta) were the subject of hacking, apparently targeted deliberately with spear phishing tactics. These hacks have been tied to Russia (albeit unclearly).
Because of this, the FBI opened a counterintelligence investigation. This was not a criminal investigation, was not focused on Trump, and was directly at the actions of Russia.

The investigation could take years, and the endgame is not always getting a criminal indictment. “The scope of this is so large,” Montoya says, “that it’s going to take a long time to not just cull through the data, but to put together the pieces of the investigation and then decide what could possibly go in a criminal direction and what could be just purely intelligence gathering.” Steinbach says, “You will try to collect as much as you can to know the totality of the threat.” Agents will ask questions such as, “Are there other actors in the United States? Who are they reporting to? What’s the flow of information? What type of information is involved?”

What muddied this up was a series of ties the Trump campaign and subsequent administration had to Russia. Most notably, it was disclosed (via unrelated intelligence work that was leaked) that Michael Flynn, whom Trump had named National Security Advisor. Flynn did some work with Russian firms and did not disclose it properly. He then lied about it to Vice President Mike Pence, who proceeded to deny the allegations in interviews. Flynn was fired as a result

Mr. Flynn, who served in the job for less than a month, said he had given “incomplete information” regarding a telephone call he had with the ambassador in late December about American sanctions against Russia, weeks before President Trump’s inauguration. Mr. Flynn previously had denied that he had any substantive conversations with Ambassador Sergey I. Kislyak, and Mr. Pence repeated that claim in television interviews as recently as this month.
The F.B.I. had been examining Mr. Flynn’s phone calls as he came under growing questions about his interactions with Russian officials and his management of the National Security Council. The blackmail risk envisioned by the Justice Department would have stemmed directly from Mr. Flynn’s attempt to cover his tracks with his bosses. The Russians knew what had been said on the call; thus, if they wanted Mr. Flynn to do something, they could have threatened to expose the lie if he refused.

At this point the FBI apparently included Flynn in its investigations. Trump didn't like this, and the press kept insinuating that he was still under investigation. So the next day Trump asked FBI Director Comey to let the probe into Flynn end

He wrote that Trump said: "'I hope you can see your way clear to letting this go, to letting Flynn go. He is a good guy. I hope you can let this go."
"I replied only that 'he is a good guy.'" Comey wrote, describing a private meeting with Trump in the Oval Office on February 14, then added: "I did not say I would 'let this go.'"
The exchange took place after a meeting between Trump and senior intelligence and homeland security officials, after which the President asked to speak to the FBI Director alone.
Comey said in his testimony that he understood the President to be requesting that he drop the investigation into Flynn, who had resigned the day before. But he says he did not understand Trump to be referring to the wider Russia investigation.
"Regardless, it was very concerning, given the FBI's role as an independent investigative agency."

Trump would later fire Comey over the investigation, assuming that Democrats (who had panned him for his statements regarding Clinton) would happily agree with it. They didn't.
So where do we stand with evidence?

There's not been any direct connection between Trump and the DNC hacks
The Russians might have hacked the DNC, and they did target voting machine vendors but there's no evidence they hacked voting machines, or altered voting counts. Recent reports indicate that voting providers were spear phished and some voter data compromised.
There is a now a special prosecutor who is investigating the matter and the Trump administration ties to Russia. It is unclear what the focus of the investigation will be beyond that.

A:

Like with many investigations, it didn't start as a Trump/Russia investigation (remember, the Clinton impeachment over consensual sexual contact that he lied about was supposed to be an investigation into a Hillary Clinton land deal from the early 1980s).
Originally, during the summer and fall of 2016, during the presidential campaign, there was evidence gathered from the intel community that the Russians were trying to meddle with the elections.  Hacking and releasing information (illegal), setting up real-ish looking news sites with phony stories that opportunistic partisans would link to as proof that their side was pure and the other evil, etc, and even probes of state and municipal voting systems, which was especially worrysome.
Initially, it seemed that it was generally Russians being jerks, as they are wont to do, and that the Dems, especially the DNC, were sloppy with their security, so the one-sided nature of what was leaked was more a matter of opportunity.
Late in the game, they found that the Russians had also breached GOP systems, but had not leaked information in a similar fashion.  This caused the intelligence organizations to conclude that the Russians were hacking not just to cause general mischief, but to throw the election in a particular direction.
Wall Street Journal (1/10/17): FBI's Comey Says Republican Data Also Hacked by Russians
At the same time, counter-intelligence monitoring had noted many extensive contacts between Trump campaign members and Russian contacts they were monitoring.  The two were not necessarily linked, but they were noted.
At the time, out of concern over the targeted nature, and particularly the probing of systems, the Obama administration wanted to make an announcement/warning to the general public.  However, Obama was worried about the appearance of partisan meddling by himself, so he called together the leaders of the House and Senate, both sides, had the intelligence people brief them, and wanted to make a joint/united bi-partisan announcement about what was going on.  Mitch McConnell, specifically, balked and stated he'd raise a partisan stink if anyone made any announcements, so nothing was said by the administration.
NY Times: Loath to Meddle in Election, Obama Delayed Blaming Russia
Flynn was the biggest problem.  He was receiving money from different foreign entities, and he lied about doing so.  He then lied during the vetting process to be Trump's appointee, and, seemingly, lied to VP Pence when stories started circulating about it. Trump was warned by acting AG Sally Yates that Flynn, especially, was compromised, but Trump pushed ahead and waved it off.  They did nothing until information was leaked about it, publicity was growing and the facts couldn't be denied.  Then they dumped Flynn, claiming they lost trust in him, but the only thing that had really changed was that the press was reporting on it.  The Trump administration's main concern was the fact that the information was getting to the press, not the conflicts within their team or gaps in their internal vetting process.
Washington Post: Flynn probably broke the law by failing to disclose foreign payments
Even so, there were separate tracks to investigations.  Unlike what many seem to be claiming, this does not mean that the investigations were unrelated, or that they would not converge, but based on what they initially knew, there were several tracks that could be followed, so they followed them, separately.  Many claim there should not be an investigation, or that it's meaningless because, currently, we don't know of direct links between parties and events.  This is disingenuous.  It's fair to say we shouldn't claim that we know something is rotten, but it's specious to say there's nothing there - we shouldn't investigate because we haven't confirmed anything.... which can't be confirmed unless it's thoroughly investigated.  If that standard would apply to all criminal investigations, our prisons would be empty.
There was the larger counter-intel investigation, which was not, initially, any kind of a criminal investigation.  This covered the broader issue of Russian involvement, and did include looking at involvement by Trump team members (not because they specifically targeted them, as such, but they would look any any kind of contacts or interactions).  Separately, because of what was already known about Flynn, those allegations were clearly of a criminal nature, so they pursued them separately.
VOA News: Several Investigations Underway on Trump-Russia Connections (3/2/2017)
As is often the case with investigations, attempts to forestall or minimize the effects of the investigation only caused them to balloon.  

Trump fired Comey, specifically, according to him, because he was unhappy about the FBI investigation dragging on
Many more members of his campaign, transition and administration team claimed not to have contact with Russian officials and have been found to have had them, including AG Sessions
Sessions said he would recuse himself from all matters involving the Russian investigation (which angered Trump greatly), but recommended Comey be fired.
Comey, after being fired, revealed that Trump had asked him to end the investigation into Flynn, which, again, was a criminal investigation, and had made memos of those meetings.
When initially riding out the firestorm of the Comey firing, Trump Tweeted vague threats to Comey.
The fact that the administration originally had an implausibly deniable reason they forwarded for Comey's firing, which was immediately undercut by Trump, led to former FBI Director Robert Mueller being brought in as an Independent Special Counsel, with a pretty much open-ended mandate to investigate all things relating to the Russian interference.
Jared Kuschner, Trump's son-in-law who has been delegated the responsibility for saving the entire planet and fixing all woes, had extensive contacts and had proposed back-channel communications with the Russian government, contrary to their own claims.  Reuters:Trump son-in-law had undisclosed contacts with Russians

Given all the matters of normal separation from conflicts of interest, and maintaining appearances and actual propriety that Trump disdains and specifically ignored for these events, Mueller immediately escalated the review of Trump involvement with trying to alter the investigation and the entire Russia counter-intel investigation (given his actions) into a full criminal investigation.
Had Trump and Co. simply kept their hands off, and kept their mouths mostly shut for the most part (the usual "don't want to comment on an investigation at this time, we're going to let them do their jobs" is rubber-stamp standard for any kind of investigation into a current administration) it's possible, if there were actual crimes or improprieties, that it would have burned them, but now, regardless, Trump's inability to handle any kind of coverage that isn't praising and laudatory has escalated the entire affair in a very rapid fashion well beyond what it normally would have been, in all likelihood.
Many of the problems being investigated are being investigated because members involved initially did not disclose contacts, nature of contacts, or lied about them. Again, frank disclosure might have raised some partisan stink, but possibly nothing that would lead to a full-fledged federal investigation, if the actual contacts were not of a nefarious nature. Lie about it, and then tell them a the real, benign story, and there is considerable doubt about the truth of the follow-up accounts (if that's all, why lie about it in the first place? Since you lied before, why shouldn't we think this is just the next level of lying?)
That's the general overview of what the case is, but the exact details are usually kept under wraps by investigators, other than leaks to the press. The actual information is often a bit different than the leaked accounts, in their fuller context, so knowing, really, the details is something we won't know for a while.  Claims that the press are making it all up are probably deflecting by those with an agenda.  Traditional, professional journalistic practices require independently verifying leads or stories with more than one source.  Claims that they are reporting the whole, "true" story are probably also overblown, at this point. Leakers often have an agenda, as well. Keep in mind that abuse of powers of office to try and steer investigations, even if the crimes are relatively benign, are considered to be a major problem.  The cover-up, or more specifically, the exercise of powers of office to facilitate a cover-up, is often a crime of much greater magnitude than the crimes or activities attempting to be covered-up. It's a lesson that has been repeated many times in recent US history. Whether it's surprising that no one learned from that history is probably an assessment that falls along partisan lines.
NOTE: I'm at work, on break, so I'll have to circle back and add a bunch of reference links, probably on Saturday since I'm also traveling today.  If you down-vote or vote to close because of this, I invite you to re-visit later in the weekend.

