I have no research experience and I'm in my junior year of college. I'm very interested in doing research in any field at this point (combinatorics or linear algebra would be especially preferred) and I fully intend to apply for math grad school during my senior year.
I have a decent amount of coursework (abstract algebra [groups, rings, fields and special topics], analysis [real, complex, functional, measures], linear algebra, topology, combinatorics, number theory, partial differential equations, and more). I want to do research this summer (even without funding if need be) and throughout my senior year (I was declined to every REU that I applied for, unfortunately) and I plan to work alone since I have no other options (that I know of).
There aren't any student projects going on at my own college (except for one on wheels) and my professors (5 total though one is leaving and another will be on sabbatical) only seem willing to help if I provide a research topic and if that topic is related to their own expertise (I don't fit in this category) which means that I've had no help so far. I don't know where to look since most journals and blogs seem to be deeply invested into very complicated projects where my ability is likely not high enough to understand. There is no graduate program at my college so there are no grad students.
My questions is this: Where can I find potential research projects or at least a start? How does one decide on what they want to do? I know people from other colleges but they were simply given projects to work on so they didn't know how to find on otherwise.
It sounds like it would be easiest to do research with one of those professors. I'm not sure why you say you don't fit in the category of one of the professors expertise when you seem to have a general range of knowledge. I would look up your professors literature, checking for what's popular and what's interesting to you. Use those ideas while looking at related papers and try to come up with a problem to explore. Often they won't want too grandiose a problem when working with students because they want you to be able to do some of the work. Dave Renfro offers good advice. In flipping through materials likely to have student works, you can get an idea of what's appropriate.
If the professors are hard to approach, show them you've done your pre-research and maybe even attempted some things. If you need help getting to that last stage, perhaps come back to SE with more detailed information, or to some mathematics forums asking for some hints/direction.