I had quarters in grad school and my MFA. the last time I had semesters was in high school and i can't possibly fathom why someone would prefer them. Obviously, I'm a little biased, but I loved the fast pace of quarters for all the reasons listed above. More variety of classes, they're over faster if you don't like them for whatever reason, etc. I also think that the quarter system really forces students (and profs I would guess) to produce material quickly, reliably and on a deadline. It took me a long time to learn that school and it's truly invaluable to have.
As far as TA advice, I'd say if you are designing assignments, make them manageable time-wise. Both for students to complete and for you to grade. Grade as quickly as possible, because other posters are right, not getting any feedback til half-way through the quarter sucks. I also think correspondance between students and TAs or Profs before the quarter starts can be really helpful. Order books early, like someone else said. Don't waste the first and last classes on typical first day/last day stuff. If your class meets twice a week, you only have 20 sessions. Make em count!