How do I start doing math research?
The easiest way to facilitate a research project in math is to simply talk with faculty members who you'd like to work with and see if they are amenable. These will likely be instructors you've had who already know you.
You might do a reading course that includes a research component (you'd design this with the faculty member). This would be for course credit as MATH1970.
You might consider applying to an REU (Research Experience for Undergrads) for the summer. These can be competitive! It makes the most sense to apply for the summer before your junior or senior year.
ICERM - here at Brown has a summer program for undergraduates!
Occasionally a faculty member might post a project to the Brown SPRINT/UTRA page. These could be for either a semester or a summer and there are opportunities for funding available on that website.
What do I need to take before doing research in math?
This is a hard question to answer because it depends upon the area of research. You might consult the Concentration Suggestions page to see what courses are expected for a particular area of interest. And if you know you're interested in working with a particular person, check with them!