Here's my two cents on these issues-
Fellowship vs. RA: In some schools, a fellowship allows you to neither do research nor serve as a teaching assistant. For example, Rochester Optics gives fellowships to all incoming students for their first year so they can focus on their studies. Lehigh typically gives TAs, but I was offered a fellowship; in the letter about it, they commented on how it would allow me to focus on my coursework. In some schools, this can also allow you time to do some "pro-bono" lab work or an extra lab rotation to gain experience, but this is not necessarily expected.
Teaching Assistantship: Based on what I've learned from the schools I looked into, a TA will never (at least at most places) teach an actual undergraduate course. Here's an example from Lehigh- first year grad students who are also TAs will "teach" a lab section which means they monitor the students and answer questions during the labs and they grade lab assignments. Second year, they will move to teach the recitation section of the intro courses. These are the sections where the huge intro undergrad course is split into smaller groups to work on homework problems. The TA is expected to teach the students how to do the homework problems. For the most part, the TA is given a lot of support on how to do this. I think the TA is also expected to grade the homeworks, but I'm not sure.
Candidacy Exams: This will depend on the school, but from what I've seen schools either have you take this after your first full year or after your first three semesters, with at least one additional chance to take it the next semester. You'll need to successfully pass it by the time you finish your first two years. I think Physics programs most commonly do the first test after three semesters (this does not include summer), but the program I'm going to in Chemical Physics (and I think Rochester Optics, too) do it after the summer of the first year.