Are you a current undergraduate? If so, then in a large university, there usually is an Office of Sponsored Projects or the equivalent that you can look up current needs for research if you don't know your own department that well. In smaller or teaching focused universities, usually there is a student placement office with faculty for those opportunities or within partnership of a larger university with a Research I-III designation.
I usually recommend in a larger university to consult with your department or advisor first for how that works locally. In a smaller university, definitely the advisor needs to be contacted as you might actually be at a teaching-only school. If you happen to be at a community college at the moment, you probably are better off spending more time volunteering. Research experience is nice but not mandatory for most places, but few applications succeed without a significant and sustained commitment to community service.