Opening a CC will noost your score in a big, big way. I recommend 2 revolving CCs and one charge card (store, gas, etc.) eventually. But you'd want to focus on that first one and getting it would be tough. Nothing is guaranteed out there. There's a couple cards that come in mind with $100 to $200 up front fees that might be a shoe-in, but I certainly wouldn't recommend that.
I'm personal to BofA as are so many others in here. They have a pre-approval selector that you could try without getting dinged with an inquiry. That is available through BofA's homepage. Also if you do apply and get denied, they'll always come back with a counter offer for either 99/500 ($99 deposit with a $500 CL) or for a fully secured card. After 9-12 months of great payments, they'll unsecure it for you and return the money. A fully secured CC is a great way to start too. After it reports, you'll see a large spike in your FICO score and that will put you in a position for unsecured, prime cards beginning a few months from now.
Whoever you choose, don't app blindly. If you get denied for a couple CCs, it might be wise to back off and find a secured CC somewhere. Getting that first one is tough. The rest is gravy.
Assuming you have no other installment debt, paying off the car could give a small ding in points. But don't go out and get another loan. Not worth the $$$ to gain 5-10 points, or whatever it is.
