It's a good idea to start doing REUs after your first year, because they are great preparation for graduate school - it's a head-start on how to do research, and you often don't get that as part of the undergraduate coursework. If you have research experience, perhaps even a conference proceeding or publication, you're ahead of the game when it comes to applying for graduate programs.
My REUs were in astrophysics, but I'm sure the same idea applies to them all. Applications are due around January, so ask your professors if they can recommend one, do an internet search, and look for REU posters in the hallways. It's a bit hard to get one after your first year (they assume you have some of the background already) so you might see if one of your own professors will take you on for the summer as a research assistant. That will put you in a good position to apply for one at another school next year.
My REUs were 2 months long, paid pretty well and usually included housing and travel. They don't expect you to know TOO much - you're still an undergrad, after all, so they'll expect you to have a lot of questions. The idea is that your REU adviser will pick a project in the field that you can reasonably finish in the two-month time period that gives you an introduction to research in that field - in my case, data analysis and modeling.