Just walk into your local bank (you do have a bank account, right?) and ask to apply for a credit card. All the banks have their own credit cards. Just apply for whatever card you want (I applied for a Platinum Card even though I was still a student at the time and didn't even have a job), and if you don't qualify, they'll give you the next best card you DO qualify for, which in my case was a "beginner's card" with a $500 limit.
On a side note, I noticed someone talking about cosigning up above. How does this work? I'm interested in getting an AMEX Blue (not for students) but I don't have a sufficient credit history. Can I use my parent's credit history to apply for a card?