The parent has a good point about getting involved in research. It is good for experience, connections, fun (you might get to work on some cool stuff, after all), etc. It is especially valuable if you intend to go to graduate school, but certainly won't hurt your prospects for getting a job with a BS.
I notice lots of responders advising that you go to this or that school. I am faculty in mechanical engineering at a top-tier university and I can tell you the following: as long as you go somewhere reputable and work hard, you should be fine. Ask 10 different people and you'll get 10 different recommendations about what is the best school to put you on a specific career path. In my opinion, the identity of the "best" school depends on the individual. Personal happiness matters as much as US News and World Report rankings. (That being said, if you're capable of getting in to a top-10 engineering program then you'd have to have a pretty big personal reason to settle for someplace ranked 100+.)
What you should do no matter where you wind up is try to get an internship with SpaceX or someone else in that industry. Internships are a great potential pathway to a full-time offer when you graduate as well as a great way for you to figure out if this is really where you want to work.
Going back to the "which college" issue, I believe SpaceX and companies like them take interns from numerous institutions. Some intern hiring has a geographic bias (because companies do not always offer relocation benefits). But I never would choose a university solely for its proximity to a particular company.
To convey my own observations: One of my graduate students interned with SpaceX this past summer (after working for Tesla Motors the summer prior). Although we are a highly reputable engineering school (ranked in top 15 by USN&WR), we're not MIT or CalTech and SpaceX isn't located where we are. Yet this didn't stop my student (and a few undergrads I know here) from landing internships with SpaceX.
The bottom line is that the school is only part of the equation. You need to work hard when you get there and it helps an awful lot to go someplace you will be happy (or else you won't want to work very hard).