While I also only completed an undergraduate degree, I did spend a few semesters doing research with a professor and ended up being co-author on a paper with him. Doing a first paper without the guidance of a researcher in the field would be very hard, but doable, and certainly performing research is doable.
I'd start with figuring out roughly what you want to investigate. Try to narrow it down as much as you can, but I understand that it'll start broad and probably eventually narrow down after going down a few rabbit holes. Eventually you have a pretty specific field you know you want to do research in. Find the top conferences in the field, look through some of the papers that seem interesting, and look at what they cite. You should be able to compile a list of textbooks and foundational articles that crop up often - read all of these. Maybe even try to replicate the results of some of the foundational papers - a lot of the time new work is extensions of the foundations, and people writing papers started out replicating those initial works anyway. If you're struggling to narrow things down or figure out what's important, focus on papers from one researcher or research group. Maybe even send them an email if you want their advice on what to read.
Finally, you have a decent grasp of the underpinnings of this narrow field and have seen some examples of how it's been extended in modern times. You might now have some ideas of what to experiment with in your own research, or if not you can try replicating more modern papers and see if that gives you ideas of what more to investigate.