I own a TI-84 (see Undo's answer), but my school provides students with TI-89s (TI-89 Titanium, specifically) for math classes, which I find to be much more useful. I used it throughout my AP Calculus AB and AP Calculus BC classes, as well as for other AP sciences. It's quite handy.
•Last for a reasonably long time (measured in weeks with ~1hr/day of medium use) on batteries, I don't want to be feeding it all the time
I recharged my school-issued TI-89 once over the course of a school year.
•Have statistical capabilities, I'm TAing an AP Statistics class this year
The specifications page quotes the TI-89 as having
Statistics plots including scatter, xy-line, box and modified box plots. Histograms and regression lines are available.
I haven't used these functions myself, but I've heard that they are helpful.
•Be usable on an AP exam. I don't want the test trolls to steal my calculator and leave me helpless on the test.
TI-89s are allowed on the exam.
•Be reasonably rugged, meaning I can throw it in my backpack without worry
I have yet to do an intentional drop test, but my classmates and I have inadvertently done many in the past. The case improves the calculator's resistance to damage.
•Be reasonably fast, meaning I'm not sitting there drumming my fingers during a test while it chews on something
The TI-89 is reasonably fast, so long as you don't throw a complicated integral at it. But even Wolfram Alpha would be hard-pressed to quickly solve something that complicated (I've tried!).
•Be within a budget, ideally < $150. Much less would be appreciated.
TI-89s are expensive, but typically a bit under $150 (see here, for example, at $142.48).
•Have a rechargeable battery, built in or otherwise
I've never used a TI-89 with rechargeable batteries, but I don't know that they won't work.
•Color display, but battery life comes first
The TI-89 does not have a color display.