Do it well for a while, apply for a position higher up when there's an opening.
Without specifics, it's hard to say anything than that general and not too helpful advice.
Check out this documentary on how to climb the corporate ladder
http://www.imdb.com/title/tt0108311/
Yeah. It's also hard to imagine that I could get anything higher up without being a full time employee.
If you can transition to that, it would help.
I'm in school. So, it's not possible at this time. When I am not in school(breaks/summers) I have the option of full-time hours. But while in school, I am capped at 19.5 a week. (It's a university job at the hospital)
Why do you care about moving up? This is a temporary position. You should be focusing on getting an internship in a software company and ditching this job as as soon as you can. An internship will probably pay more too ($20+/hr).
QFT.
The $20+/hr internships are usually cs tough. Mine were $14-$18.
Honestly, it's people skills and networking skills on the job.
Yeah, CS and EE internships pay more than most. A mediocre CS internship would pay $20/hr. A good one would pay $25-30. Some are even $35+.