A dealer would buy your car if they think they can flip it on the lot within 30 days. If not, they'd plan to just sell it at auction, meaning you're going to get whatever they think it will sell for by sight only (they have a database of what cars are worth at auction) less the $1,500 they're going to want to make on the deal. In other words, if you're not buying your new car from the dealer, you can expect to get hosed if you sell it to them at all. If they even want to buy it at all.
Best bet is a private party sale. You'd want to research the value of the car. I'd recommend meeting in a neutral location and not at your house (if there are problems of any sort after the fact, you don't need anybody knocking on your door). You'll want to figure out what you need to do to exchange title (if you owe money on the car, the dealer holds the title and the bank that lent you the money has a first-postiion lien on it).
My take-it-completely-at-your-own-risk advice is to sell it private party. See if you can't find a mechanic with decent business who will (for a fee) let you park it at his location and try to sell if for you. Those guys always know people in the market, and you can always list the car and meet people at the mechanic's. It also lends a little bit of credibility to your sale since you've got the car at a mechanic's place of business and not in your own private driveway.