EpiJunky has a must-read post for anyone starting out in EMS.
The only thing I'd add to that list would be this: Build, and foster, a passion for learning new things.
I had a good paramedic program, and my instructor was fantastic (Thanks, Dawn!), but I've learned more about medicine out of school than in. I learned from talking to doctors, nurses, and specialty technicians (respiratory therapists, etc.)
I learned from reading books... Taber's Cyclopedia, Merck Manual, and Tintinalli's should be part of your library. A high-fiber diet and comfy toilet seat help promote reading.
I learned from listening to speakers. I learned from debates with EMTs and Paramedics that were much older than me. I learned from the ones that were younger, too.
Every patient you meet is a chance to learn something. It might not be an obvious lesson... you might have to dig around a bit to find out what you can learn, but it's there. Go looking, and eventually you'll be a better EMT for it.
Learn. It'll be the best thing you can ever do as an EMT.