Should I put an ongoing research position that I recently started on my resume? I started doing part time undergraduate research under a professor last week. I have a general understanding of what I'm working on, but I do not have an explicitly-defined project (rather, week-to-week my work will be decided based on the previous week's progress and what work I can reasonably do).

Today is the start of my school's application process for co-op. I would like to put this position as ongoing employment experience on my resume, but I am not sure how much I should put on there.

I am hesitant to ask the professor I'm working with because I'm only a week in and I don't want to unintentionally make him think that I'm just doing this to pad 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!