Should I put an ongoing research position that I recently started on my resume?
It's up to you, but the guiding principle is "Anything on my resume is fair game for the interviewer". If you've worked long enough to be able to talk for a couple minutes about what you do there and what you've learned so far, then go ahead and put it down. If you wouldn't feel comfortable talking about it, leave it out.

Resumes do not need to have a complete breakdown of every job position you ever had; you're highlighting your strengths to get the recruiter/manager/professor to notice you, and ask you questions on the interview. The resume is mostly only good to get you to the interview, so again, only include it if it something you can talk about comfortably. Good luck!