Don't go to Caltech for aerospace unless you just want to learn airplanes. I work at JPL and have a lot of interaction with CalTech students who complain about the lack of space stuff in the aero department... they don't even have an orbit mechanics class. MIT is ok if you want to do systems engineering, but generally their aero department doesn't do much space stuff either (last I heard, their orbit class was taught by a grad student who took it upon himself to have some sort of orbit class).
If you want to do SpaceX, I'd write them an email and ask for their advise, ask where they recruit from. They will probably want chemical prop and systems engineering people.
From what I've seen the best schools if you want to do space are Carnegie Mellon, Purdue, Colorado, UT Austin, Georgia Tech, Stanford, Michigan, UCLA, Cal Poly San Luis Obispo. Carnegie Mellon and Stanford have awesome robotics programs. Michigan and Caly Poly SLO have excellent cube sat programs. Michigan, Stanford, and UCLA have excellent electric propulsion. Georgia Tech and Michigan have excellent systems engineering. Purdue, UT Austin, and Colorado have excellent orbit mechanics. And Purdue has probably the best chemical propulsion program. Georgia Tech has a really amazing senior design class (best out of the 5 that I've advised as an industry person).
If you don't want to go to far, I'd recommend Michigan, Purdue, or CMU. But try to email SpaceX and see what they advise (but be aware that the person who responds will be biased towards their alma mater)