Q:

How do I get rich quick?

I haven't played Terraria in a while and it has changed a lot, so I want to start over again. What should I do to get some good equipment as quick as possible?
I don't need to have the epicest equipment available, but I'd like to be able to roam the world freely, mine where I want, and have some cash. 
(I only have a little previous experience with the game. I mostly dug down, found a cave and explored it. Dug even deeper, which took quite a lot of time, found a big cave, died a lot, was too weak, died some more and then moved on to other games.)

A:

I think the best solution in the early game is to explore your starting biome and search for shallow caves that contain treasure.  You'll likely find a boomerang early, which can be useful as a ranged and indirect fire weapon early in the game.  
Once you've come across a few chests and have a bit of gear/weapons, start digging a shaft.  Explore caves and fight enemies, which will probably yield some amount of cash plus opportunities to mine ores you can use to forge better equipment.  Better equipment will allow you to dig deeper and/or explore some of the neighboring biomes, which will yield opportunities for even better gear.  
Money's not super important as most of the gear in the game can be crafted or found as treasure.  You will need to purchase items from time to time, so make sure that you occasionally make time to satisfy the conditions for NPCs to arrive.
Lather, rinse, and repeat until you're epic and difficult to kill, and then you can tackle some of the more exciting boss fights and deepest regions of the world.  If you're at a loss as to what to do next, or how to craft some of the higher level gear, go dig around a bit at one of the Terraria Wiki sites, which contain info about all sorts of interesting artifacts.  
Death is relatively cheap, as long as you leave as much of your money as possible at home.  Don't be afraid to go and explore something new, you might luck out and find a valuable artifact.  The worst case scenario is that you end up a tasty snack for something big and nasty, and maybe lose a few coins.  
The goal of this game varies depending on what your goal is, so don't be afraid to do something risky or try something new if you're having fun!  If the game is getting frustrating, try exploring more closer to the surface, or looking up a strategy for dealing with the particular situation you're stuck on.

A:

The fastest way to get what you want is to rush to Eater of Worlds & a set of Demonite equipment.
I've found the fastest way to do that, starting from scratch, is to obtain the following

At least two pieces of silver armor, or better
A bow & arrows. Copper is good enough but better will make things faster.
Ironskin potion. This is the key element - it's frankly overpowered.
Regeneration potion. Not as strong as Ironskin but very easy to craft.
A stack of Shuriken

Then you kill the Eye of Cthulu, which drops demonite ore. With demonite alone you can only make the axe, bow & sword.
What you really want is the Dryad, who sells purification powder. This lets you dig through ebonstone and smash shadow orbs. Smashing shadow orbs lets you fight the Eater of Worlds and obtain Shadow Scales, along with even more demonite. (Smashing three orbs is faster than farming for worm food.) Before you venture into the corruption chasms, be sure to obtain a hookshot. When you fight Eater of Worlds for the first few times, use an Ironskin potion. I find that grenades slay Eater very quickly, and they're reasonably accessible.
Summoning Eater a few times through orb busting will net you the highest defense armor short of Hellstone; a pickaxe that can mine every pre-hardmode block; a fair quality sword, axe, bow, and hammer; the musket & arms dealer; and some nifty trinkets.
A full set of Demonite gives enough base defense to farm Eye and Eater without buff potions, and pre-hardmode I am aware of no faster way to obtain coin than farming these bosses & selling their drops. You will also activate the goblin invasion, which you must defeat once for the tinker and mechanic.
In some worlds, the crimson replaces the corruption, which also replaces ebonstone with crimstone, demonite with crimtane, and the Eater of Worlds with Cthulu's Brain. None of these substitutions breaks the overall strategy.

A:

Normally I sell my items for money. You should gather tons of life crystals. 1 life crystal is worth 1 gold and 40 silver coins. I got at least 76 gold coins.
BTW, this works for iOS/android, console, and PC. I tried it on all versions just to make sure.

