Bad credit, need a car

Income - $3400/mo (two paychecks/mo, $5100 when 3, effectively $3700)

Rent/Utilities - $820

Student Loan - $550

Food/Phone/Gym/Etc - ~$650 (Rounded up and added $100 to be safe)

So effectively $2k/mo in expenses currently and I'm looking to add a car. I've currently got $3k saved up.

Two problems:

1. Credit Score - 600

2. Reason why, I got laid off and struggled with depression for a year+ and my Private loans are in charge off. I'm working on creating a payment plan to get those settled.

Outside of that, my credit history is pristine. While unemployed I had everyone work with me so that I wasn't under too much (except my private student loans, they just shrugged and did nothing). I'm completely caught up on (and ahead of) Federal loan payments, have no money owed to credit, finally have a substantial income again (almost 2x what I was making before I was laid off).

I need a car for work (I will start traveling in a month's time).

I've been looking at cars in the 10k-13k range, newer mid size cars (I'm a tall person, compacts don't work). I'd like to put $3k down, and then blow through the loan payments as fast as possible so interest doesn't gouge me too badly.

Should I be looking for financing through dealership or another source? What sorts of institutions would provide me with the best rates?

Edit: I am aware I can pay $3k cash for a car, but my plan is 10k-13k with 3k down.