What steps should I follow in order to be able to start conducting research if there's no sign of support within my department? My understanding of research is that it's a way to find solutions/answers to questions/problems in a certain subfield of a bigger discipline, which in my case would be Computer Science (honestly, it'd be more like Computer Engineering).

I think I know, however, that if these problems and solutions are too simple or banal, the whole process of doing research is irrelevant because 


surely somebody else has already worked in that, and 
there's no sense in working in something that won't contribute a significant amount of knowledge to the discipline. 


This means that in order to get worthy (or at least relevant) problems to research, you must be at the cutting edge of the field so that you have an idea of what's not yet known and you can start working on it. Am I right?

If I am, that sounds kind of complicated. I have heard that I should read lots of papers in an area of my interest and work from there, but I'm sincerely not sure about how to start doing even this. I have access to the ACM Digital Library, for example, but I don't know what to search for, what to read, or in what order to do so. 

I am alone in this, for my university is weak in research and no professors seem to be interested in it. Do you have any tips, or any specific set of steps that you followed to get into research?
It'll certainly be easier if you can find a professor willing to supervise you, so I strongly suggest that you keep trying on that front. But even if you can't, all is not lost! Here are the steps I recommend. Note that this may not be the only way to get started, but it's a way that works for many of my students:

Step 1: Start by identifying an area of interest

A good way to start is to identify a topic in class that interests you. Approach the professor for that class and ask him/her to suggest some related papers for further reading.

Step 2: Get a better view of this area

Thoroughly reading those papers may take a while, since this is new to you. Don't be discouraged by this! After you have a reasonable understanding of the papers suggested by your professor, start to branch out: look up the papers cited by/citing those papers to get a broader view.

Step 3: Reproduce some existing results

Somewhere in all this reading, you'll come across something that really interests you: a proof, a design of a system, a software implementation of an idea. See if you can reproduce this result (by going through the steps of the proof, writing your own software, running a simulation, etc.) Again, this may be more difficult than you expect; don't get discouraged.

Step 4: Do something new

Now that you've reproduced someone else's results, can you extend them? Is there a small variation on the system design that could improve the results, an interesting application in another domain, a stronger result you can prove? Get to work. 

Step 5: Communicate your results

Congrats, you are now almost a bona fide researcher! Go show the professor who recommended the papers what you've been working on. A big part of doing research is communicating the results, once you have something to show. (And to get into a good grad school, you'll need to impress some potential letter writers.)

You should also consider writing a draft of a paper and soliciting feedback on it.