Question: I am interested in getting a credit card but have essentially no credit history because I had a co-signer on my college and car loans. How can I build my credit if no one will give me a credit card?
Answer: I'll get to your answer in a second, but first, I'm skeptical that no one will give you a credit card. I mean, you do have a credit history. When you have someone co-sign your loan, that's part of your credit history. As long as you didn't completely botch things up, and your car wasn't repossessed and you're not 19 months late on your student loans, then, well, you shouldn't have too much of a problem getting a credit card. Of course, I say that not knowing your situation. Maybe you aren't employed, for instance. That can be a roadblock to getting a credit card.
But chances are, if you have a source of income and you apply for the right credit card, you'll be accepted. Now, you may not have the excellent credit record required for a premium rewards credit card, but many gas cards or low-interest credit cards with no major frills and perks, should be available to you if you have a credit history, even if it's a shared history with a co-signer, of paying down those student loans and car loans on time.
Now if you've already applied for a card or two, and you've been rejected, be sure to find out why. If you have a spotty credit history, figure out what you can do to clean it up a bit. Then you may want to apply for a secured credit card. That's where you put your own money on the card, and for about a year, if you use the card without any problems, and you make all your payments on time, then you'll get the money returned that's still on the card, and you should be able to transfer to an unsecured credit card.
And finally, if nobody actually will give you a credit card, and you don't want to get a prepaid credit card (which will do nothing to help you build credit; some people just like the convenience of having one), then just keep paying your bills on time. While having a credit card is one of the fastest ways to build up credit, one of the best ways is to simply pay your bills on time--whatever those may be. Be financially responsible long enough, and eventually credit card issuers are going to notice.