I'm not sure if you would be in a position to do the US Bank card. Chase might require at least 6+ months of revolving history before you are approved for a card. I would check your credit reports to see how and where the car loan is being reported. Annualcreditreport.com allows you to pull all three once a year for free. The cards you need to target should be student cards, you're in a great position in that you can get halfway decent rewards with more lenient underwriting standards. Take advantage of it.
Also as along as you are planning on paying your bills in full, 0% is not a necessary requirement. You only get charged interest if you carry a balance from one month to the next. Check out creditboards.com for further information on building your credit.
Good luck and welcome to FT!