Always focus on the sale price, and nothing else
Down payments, monthly payments, and interest rates can wait. Your trade-in isn't going anywhere, either. You need to negotiate the price of the car based on what the car is worth, not what you can spend. Opening with, "I've got $3,000 to put down and I can afford $300 per month," isn't just bringing a knife to a gun fight, it's bringing the gun, too, then handing it to the dealer, who will use it against you.
Beware the "let me take this call in another room" trick
There's an urban legend that cubicles at dealerships are bugged. By and large that's false. But at some dealerships, a colleague will call the person you're working with, so he or she can pick up the phone, "take the call in another room," leave the speakerphone on mute, and hear you discuss the options with whomever you're with. You don't want the dealer to know your absolute bottom line, but the dealer really wants that information, and will sometimes resort to dirty tricks to get it.