Calculate what a reasonable value is for the vehicle you're buying. Then look at the trade-in value of your vehicle. Figure out the difference and then focus on that number when you go to negotiate. A dealer can tell you they're giving you any amt for your trade but the important number is the difference. And don't let them talk to you in terms of the payment amt. You don't know how long you'll be making those low payments and at what interest rate. Save the payment talk for after you've negotiated the amt required to trade.
And I guarantee no matter what you have to trade, they'll tell you the mkt for that vehicle is crap. Of course people are beating their door down for the car you're interested in, so you better move quick.
Block off a big chunk of time to negotiate. It can be a war of attrition. Don't be afraid to stand up and take a step towards the door. They can't sell a car to someone who walks. And if you do walk and they call in a day and ask if you're ready to buy, ask them if they're now ready to accept your offer.
And if they won't deal, forget 'em. My dad said there's always another car around the corner. I've found this to be true about women, too.