First of all, my engineering school is better than your engineering school
now that we got that out of the way, let me try to give you some advice!
Internships are always a good idea, but keep in mind, when you work for an auto company, it will also be a 'big name' company...
One thing to realize about working for an auto company is that you will most likely be working on emissions systems, so if that is what you want, cool. I choose to keep my job and my interest separate, when I want to do car stuff, I work on my car.
A true, hands on engineer, is most likely gonna be on a factory floor, working QA or something like that. I worked on machine validation at Goodyear in Kansas, very hands on, no desk.
You can PM me your resume if you would like, I don't mind helping a bother-in-trade.