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It's always a good idea to get a couple more cards to round out/ thicken your credit profile. I think your best rates will come from dealer special financing or a CU. Sometimes a dealer will get you a better rate from your bank then you can get from them directly. The biggest problem you have is no previous auto experience this can result in a higher rate. If you use the 5k down payment that can help lower it a bit as you have skins in the game and the bank will be close to used car price on the new car so less risk for them in case of default.
Maybe get a CU preapproval before going to the dealership and make them beat it (this will also tell you what to expect if the CU quotes you a high or low APR).
Keep us posted on how it turns out!
Not an expert on credit but I have some real inside knowledge on applying for auto loans. Based on your situation, your are more than likely to to be approved in a lower single digit apr. The only piece of the puzzle missing is your income, that will either make or break from what I see.