If you are up for a challenge, look into MANOVA (multivariate analysis of variance) and its assumptions. It would fit the case where you have multiple categorical predictors and multiple continuous outcomes. In essence, MANOVA would help you describe the web of relationships connecting your predictors to your outcomes.
If you want to keep things simpler, you can analyze the height-weight relationship separately using a scatterplot and (if you find a linear relationship) correlation. For your main research questions you could conduct separate ANOVAs (analysis of variance) for height and for weight. Again, you'll want to look into the assumptions underlying best-practice use of ANOVA.
Depending on your version of SPSS, for modeling you'll want to go into Analyze...General Linear Model...[Univariate or Multivariate]. The Help files there should prove somewhat useful, and if you run into a link saying "Show me," it should take you to a tutorial; most of these are pretty good.
For the scatterplot, type graph/scatter height with weight.
For correlation, it's simply corr height weight.