I have a few friends who did research between first and second year. You're right, it's not really reasonable to expect students to do research during first and second year when we have a lot of other obligations. You could possibly continue research into second year if you started a project in the summer if your project was pretty low maintenance, but it seems to me that most students who did research over the summer didn't continue the work but did have to present at a poster session during the year. You can often apply for a grant/fellowship that includes a stipend for summer research, but of course this is not going to be a ton of money. People do it more for the experience rather than for the money.
It is possible to get time during 3rd and 4th year to do research, but I think this depends a lot on your priorities, your school and your schedule. What residencies are you interested in? Research is generally a good thing, but I think it only works for you if the particular residency values research and you value research too. It's not worth taking 8 weeks out of your rotations to do research if it's not something your interested in and you don't plan to pursue research in your life after med school. It's more important that you have good clinical grades and letters of recommendation.