Well, I honestly can't recommend that you finance the car, but with that said, the first hurdle is getting financing on a 10 year old car. Lots of places have mileage and age limits on loans they'll give you, and 10 years is right on the edge.
Assuming that isn't a concern, there are two ways to do financing. You can go to a bank/credit union and be pre-approved for a certain amount (loaned money is good for either private party or dealer sale), or you can finance through a dealer (loaned money is only good with that particular dealer). If you get the loan through a bank/credit union, when you buy a car at a dealer, it is similar to buying with cash, other than additional paperwork, the dealer gets whatever amount is agreed to.
If you have to finance, financing through a bank/credit union is generally preferable than to go through the dealer. The bank/credit union will generally have better rates and better terms. This will also give you more flexibility to cross shop between different sellers. Between banks and credit unions, credit unions tend to offer slightly better rates and terms, though that really depends on each individual institution, and you'll have to research and cross-shop. Financing through the dealer I think tends to be a little easier (ie. they'll loan to people with worse credit), but you get worse rates, and the loan is only good at that dealer or dealer network.
As with anything in buying a car, everything can be negotiated, even the financing and terms. Just because you're financing does not lock you out of being able to negotiate for a lower price. However, if only one dealer will give you financing, and you desperately want the car, you may end up being locked into that deal just purely by your emotional desire, which means you'll lose one of the most powerful negotiating tool: your ability to walk away from a deal.
All this financing talk though, is contingent upon being approved. Not having any credit history makes it incredibly difficult, especially with banks/credit unions post-crash. Not having credit history is almost even worse than having a bad history, because then at least they know how bad you may be. I'll be honest, I think even with a co-signer, it's going to be difficult for you to get a loan for a 10 year old Lexus, at least at any sort of decent rates from a bank/credit union.
Just another tidbit, if you haven't already thought about it already, you should go get a credit card from your local bank, just to build credit. They'll start you off with a small limit, use only about 1/3 of the limit during each billing cycle and always have enough cash on hand to cover your credit card balance, and make sure you pay it off in full every month. You have to start building credit.
(I'm guessing it's due to the safety features the Lexus has over the Mazda)
