Another "which tests to use?" question

Hi all, longtime lurker, first-time poster here. Reading people's questions and answers here really helped my understanding back when I was taking stats in college (not a math major), now I'm starting research in grad school and they've given me SPSS and said OK, start analyzing this data, all our statisticians are busy so you get to do it. Guess it's one of those "throw 'em in the deep end and they'll learn to swim" sort of things.

Anyways, I need to know which test(s) to use on the following scenario. It's fictional and kind of goofy, but it mirrors the actual work I'm analyzing so the interpretation should be the same.

- 35 subjects across 3 racial groups: 12 Caucasian, 11 Asian, 12 Black
- Skin thickness in mm measured at 3 points on each subject's body: palm, elbow, ankle - Then, apply drug previously proven to increase skin thickness at each point on each subject & remeasure thickness on those same subjects a few weeks later (i.e. two time points, predrug and postdrug)
- Assume population normality & random sampling

And our goals:
#1: Compare mean pre-drug thickness at each point between each group, i.e. mean predrug Caucasian palm vs mean predrug Black palm vs mean predrug Asian palm, then same for elbow, then same for ankle ("does the predrug thickness vary significantly between races?")
#2: Same as the #1 but for post-drug thickness ("does the postdrug thickness vary significantly between races?")
#3: Within each group, compare pre-drug thickness at each point to post-drug thickness at each point ("does the drug indeed significantly affect skin thickness as previous research has shown?")

Kneejerk thought is to try some sort of ANOVA approach, since it's clear we have more than two groups here (duh). I also know we can do repeated-measures stuff when comparing predrug to postdrug since the same 35 subjects were measured both times. Is it really just as simple as:

#1: a two-way ANOVA for predrug data between races
#2: a two-way ANOVA for postdrug data between races
#3: three separate two-way repeated-measures ANOVAs for predrug vs postdrug data, one test for each racial group?

...with all ANOVAs including a priori tests (e.g. sphericity) and post hoc analyses (e.g. LSD, Tukey) as necessary.

If I can at least understand the method I should hopefully be able to figure the rest out and also how to make SPSS do what I want it to. Thanks much in advance for your help.