As an undergrad, a lot of "research experience" is working with a prof/their grad students entering data, running subjects, reading papers, discussing them in lab meetings, coding, maybe some data analysis, lit reviews, etc. That is pretty much what I do.
A research methods course usually lets you learn about analysis and my research methods course was set up so that we had to design and carry out a project from beginning to end in a group:
think of hypothesis, design methods, write all the papers for the IRB, collect data with participants, enter data, analyze data, write a full APA style paper, make a poster, and present it at the student poster presentation night. And there's a chance it ends up as a poster at a real conference.
Then I am doing that next year (senior year) as an independent study/thesis..except by myself...again, with the hopes of it ending up as a real poster or something.
I think that's really the most research you can get as an undergrad unless you get a start on it really early working with a prof and end up with a publication--but that's rare.
As for MA programs, the ones I have looked at all weigh research experience heavily. I would still try to work in a professor's lab so that you can have some experience.