It's hard to say what works best since we have so many different backgrounds on the team. I think having an interesting portfolio of projects (professional or side / hobby type stuff) is important - make sure you've done some awesome stuff that you can map onto what you might do here. Also, hone your communication skills. A big part of success at SpaceX is being able to work with others in a team and quickly get ideas across. Getting an internship is definitely a good way to go - we usually give interns pretty meaty projects that are a good taste of what full-time work might be like. Good luck!

My recommendation would be to consider adding CS as minor or double major. Then try and get some internship and/or on campus lab experience doing software or robotics work (robotics software is pretty close to flight software). Another great way to get experience is to get involved with extra-curricular group doing some kind of robotic or aerospace challenge. Think DARPA grand challenge, NASA Sample Return Robot challenge, or one of AUVSI's robotics competitions.

SpaceX has an intership program, that might be a good thing to look into.