Ailwee Caves - fee to go on a guided tour of the cave. No fee to park. There is a gift shop (with fabulous capes!) and a small restaurant. Very very dark caves!
Newtown Castle - fee to enter the castle. Minimal. If you want one of the family to guide you, it'll cost a little bit more. - no fee to park.
Cliffs of Moher - no fee to park or walk to the O'Briens tower or to enjoy the view. It will cost one Euro to climb to the top of the tower where there is a telescope that will cost a bit too. There's also a gift shop where you can spend loads if you want.
The last few times I have been to Quin Abbey it has been locked up in the spring and autumn. It probably will be in February too. There is supposed to be a small admission fee but since it most likely is not open, you won't have to worry about it. You can walk around the abbey and visit the cemetery. Parking free.
Cliffs of Moher. It is free to visit the cliffs, however, in order to exit the parking lot you must pay a fee. Last time I was there is was about 5 euro. Technically it is not a "parking" fee but an exit fee. So they get you going! A fine Irish joke if I ever heard one. Small fee to go up in O'Brien's Tower. NOTE: Lots of construction going on at the cliffs as they build a new visitor's center. If you visit late in the afternoon or evening after the parking lot attendant leaves then you don't have to pay the parking fee.
If you're worried about fees to various sites then check out getting a Heritage pass (Duchas). I believe it costs around 20 euros and will give you either free or greatly discounted entrance to a lot of sites.