How do I repaint a bathroom that was originally painted with oil (possible lead) then latex?
You should be able to test the paint for lead content using one of the lead-test kits you can pick up at Home Depot or Walmart. Especially if the oil-based sub-layer is exposed in places. If the first test proves negative it's worth scuffing the paint with some sandpaper and testing again, just to be sure.

If it's lead-free you're home free.

Depending on where you live, and how much area you are doing, there are homeowner lead-paint removal classes that show you how to do the job safely. Not all cities allow this however.