Anonymous Coward,
First of all, don't be afraid to post who you are around here. People might make a joke but so what.
Moving on, I too am a recent graduate with a degree in Computer Engineering. I graduated in 2005 from a school with a strong engineering reputation. While searching for a job out of college, nobody wanted an entry level engineer. In desperate need of a job I took a first line support gig with the intention it would only be temporary.
After a year at that job I realized that the promises made to me upon hiring of moving to engineering were not going to come true. In addition I had a bad feeling about the company as a whole. The company was sold three weeks after I left for a cool 6.2 Billion Dollars.
I managed to land in another tech support job, but this one is quite different. Instead of dealing with grandma and her stupid UPS beeping, I was dealing with enterprise software. Oracle, WebLogic, etc. For a starting salary of 50K in a support gig, I was willing to do it again. Now a little over two years later I have expanded my knowledge of Java by troubleshooting issues in Struts. I have been able to improve on relational databases using Oracle. I've already been promoted once after one year and I'm up for another shortly. My salary has increased over 50% since my start at this company.
In addition I am taking a class on Ruby scripting and starting my own web development project on the side to build my development resume. Lastly I'll be applying to grad school as well.
My point here is not to put a handprint on my back. It is to acknowledge that tech support sucks completely. The only way out is to distinguish yourself.
I hope I've helped, even a little.