You could sell it for it's current value especially if it'll pay it off and with whatever is left over buy yourself a used Toyota or Honda like a used Honda Fit of some sort. Obviously have a mechanic check out any used vehicle you would look to buy of course and make sure it's not recked, flooded or stolen. Even if the Honda is significantly older it'll probably run better than a Fiat and be a more dependable source of transportation. Hopefully with that you'd also have cleared your car debt as well. 

Just get rid of that stupid thing, they're aren't known for longevity.

A problem will be that while in decent condition it's probably worth at least what you owe on it, the bad transmission puts you pretty badly upside-down. If the problem is not very noticeable yet and no check engine light you might see what a popular dumping ground for bad cars like Carmax would give you for it. If anywhere near what you owe I'd get rid of that thing in a heartbeat before it completely craters on you.

Sell the dadgum car! If you are upside-down (which I doubt is the case), pay off the difference between what it is worth and how much you owe. Then sell the car and pay for the new car with a cash deal with the money you got from the Fiat.