The first step is to read about and understand a topic. If you were an undergrad and you asked a professor for a research problem, he'd throw some papers and books at you and tell you to get lost for 2 weeks (or a month, or a semester, or a year). So if you wanted to go about doing it yourself, you'll want to pick your problems wisely since you don't want to bite off more than you can chew, especially if you have no guidance from someone who has done it all before.

Thinking about it now, it will probably be difficult for most to do research completely on your own. It will probably take you much longer than it would if you had some guidance. The upshot to it is that your understanding is yours alone, and if you don't understand something then you should be working as hard as you can to alleviate that. And that work is what might make you better than peers who had advisors that helped them a lot. Not that one is better than another, or anything like that. Different strokes you know.

If I were you, I'd try to contact some professors around your area, or failing that, just some professors or researchers anywhere. Ask them about their problem, maybe visit them and have a chat, and ask if there is a small project that you could work on. Researchers always have a million ideas, but they don't have the time to figure out which ones suck. They will probably be happy to have some free labor, and you'll gain your experience and maybe a recommendation letter too.

There are some things you can do that are not really, truly considered core research, but would help you when you are doing research and would be good to know. For example, contributing to the Scientific Python project or an open-source CFD or astrophysics code would help you to understand how people write these codes and how to make your own contribution. Maybe there is a method not yet implemented in SciPy, or there is a newer algorithm that makes some operation faster. You can get the publication about it and translate it into fast Python code and contribute it. That's a really cool and easy way to understand research better. Most of research is just reading and understanding, so if you can read a paper and then do some real things with it and show that it works, that is making progress.

It might be easier to suggest more specific topics if you mentioned what your interests are. CS is pretty broad.