Can't comment on where to get your licence's/ratings, I'm just about to sit my PPL test as well so I'm not a great fountain of knowlege, but I can tell you that (a) at my school if the instructors know that they're going to have a fairly tight turn between a TIF and a student later they'll preflight the A/C before a TIF student arrives so that they can get away relatively quickly and (b) the shimmy is normal as well if you haven't yet slowed down enough - if it continued to be uncomforatble in the taxi I'd then be worried. ;D
Glad to hear you enjoyed your first flying lesson, and good luck, sounds like you've got some big decisions to make.
~ QF5