Financing to improve credit

I need some advice. I'm planning on buying a car in the next 2-3 months and I'm trying to do as much research as I can before I go look at any cars. I've only ever bought a car once and I paid in full so all this is new to me.

Both me and my husband have very bad credit. We're working on it and it will hopefully improve before we buy a car, but for now I'm going to assume it won't so I'm prepared. The good news is we will have the ability to make a large down payment. My question is, how big of a down payment would I need to make in order to have a low interest rate? I will actually be able to pay in full, but I was hoping making car payments would help improve my credit. I'm just not sure how to decide on the down payment and monthly payments since I don't trust dealers to give me honest advice. I read something about hiring a car finance broker, is that something I should look into?

I know I don't want to lease, I don't want a new car, and I'm looking for under 10k.