What is the best way to go about doing research as an undergrad (pointed question inside)?
There is no harm in asking.

To answer your tl;dr question, it depends.  "Research" is not a monolithic thing.  Some professors may be working on projects that would be accessible to someone without a strong background in the subject, or would be willing to work with you to bring you up to speed.  Other projects may need someone who already has a strong background in a particular area and can jump right in.  

Clearly you would be looking for a project of the first kind, and the only way to know whether anyone has one to suggest is to ask.  If you hear of something promising, talk to the professor; try to get a sense of whether it seems like something that would be interesting and accessible to you.  Be honest with the professor about what background you do or don't have.  (Don't try to present yourself as knowing more than you do; if you end up working on a project that you aren't prepared for, the result will only be frustration for everyone.)

If you find something that seems to be a good fit for you, great; if not, try again next year, with more courses under your belt.