New vs used car buying - how do I figure out the most cost effective finance option?

This will be the first time I've had to finance a car (I've always paid for cheap cars in cash) so I don't know where to begin. What steps do I take and in what order? There's a million resources out there - which of them are useful and reliable? How do I avoid a bunch of credit checks that'll mess up my credit?

I'm primarily orienting around monthly payment, which I'd like to be under $300. I can afford to put about $7,500 down. I hear generally that new car finance and insurance rates are lower than used car, but of course a used car is probably going to have a lower sticker price and insurable value. With such a limited ability to make a monthly payment, and such a small down-payment, can I even find a new car in that range?