It's a tough question, not so much what you can get out of your old car, but how much the time costs when you take the effort to sell your old car. The only person who can determine that value is you.
Dealers love it when you trade-in your old vehicle. Why? Because if they take it, they are only going to offer wholesale price for it; wholesale price is lower than market prices, in most cases. Dealerships exist to make a buck. They can't do this if they're buying your car at great prices.
The silver lining to trading in your vehicle is that the dealers handle all the paperwork and you get away conveniently with a new car in tow, very little paperwork and very little hassle. There is a hidden benefit to trading-in to dealers though. In many states, the sales tax is calculated on the difference in price of your new car with the amount the dealer gives you for the trade-in factored in. Of course, this doesn't apply to everybody. Check with your local state regulations to see if this applies.
Do not become obsessed with the trade-in value of your car if you do decide to trade-in to a dealer. Once the dealer realizes that you're looking for a specific trade-in price, he can easily offer you that amount, but jack up the new car's prices to match. You won't be getting away with a bargain like you'd assume.
Selling privately will probably net you more cash upfront. However, you have to deal with the hassle of having potential buyers coming over to check out your car, as well as the paperwork, and so on. If you don't mind holding on to your old car for another week or two, then it might be a better idea to sell the old car yourself.
Regardless of trading-in your car or selling it privately, you should probably get all the usual routine maintenance done right before you try to get rid of the old car. A detailing and vacuum would not go amiss.
Private buyers and dealers will be sure to check the car's exterior and the condition of the engine and fluids. It would be a good idea to have a detailed record of the car's service history handy to prove the car has been consistently serviced.
Do research beforehand and set clear goals of what you want for trade-in value, what you're willing to spend, and what your expectations are for financing. The key is to hit those goals as close as possible.