Theoretically you should not be able to just trade cars at no cost. ALL things being equal, at a minimum you will pay some sort of loan origination fee, registration on the new car, likely a document fee on the new car and sales tax depending on your location. Your goal will be minimizing your costs via properly negotiating the deal.
The trick about buying cars is really about the stages of the negotiation. You have 4 basic pieces, the price of car you want, the value of the car you have, the remaining loan balance on the car you have, the cost of the money for the car you want. You want to make sure these four things are talked about in an order that benefits you.
First, get yourself the best possible estimate of the value of your current car. Go to carmax or something like it and get it appraised. Then you know your cash out number. Obviously, compare this number to the remaining balance of your existing loan to determine whether or not you're under water.
Second, identify the car you want and start checking them out at dealerships etc. When you're ready to buy, they'll try to talk to you about a loan payment. Get them to talk about the price of the car.
Third, once you've settled on a price for the new car, settle on the loan terms for the new car. You might be well served by getting yourself preapproved at a local credit union, this is an easy bluff if you simply know the credit union's prevailing rates and a realistic idea of your credit score.
Fourth, tell them you want to trade in your current car. Since you've efficiently made them sharpen their pencils on the price of their car, they'll give you a lowball for your trade in, but you have your carmax offer to counter with.
There are too many pieces to determine whether or not you'd be able to trade cars for no gain or loss. Financing a used private party car can be tricky so you'll likely achieve the best results at a dealership.