Timesheets:

A) If you have a fellowship, I believe Grad Division handles the disbursements. B) usually during the semester if you are a GSI or GSR, you just want to fill out timesheets for the whole semester (and have your PI/Instructor/etc sign them) and turn them in to the accounting people - whether you do this or not, you will still get paid on the 1st (or before, sometimes, if it falls on a weekend), but they want the records.  and C) working during the summer, you will probably have to fill out an actual timesheet of hours worked, get it signed, and turn it in before a specific day of the month (check this when you first get employed for the summer/semester) in order to get paid on time.

I think it's good advice to always check with the HR people to make sure there isn't anything you need to sign or anything else like that when you switch your appointment from GSR to GSI or from academic year to summer.  It's a good way to make sure your paycheck keeps coming in without interruption.