Drive from Palermo to Taormina

Taormina is both very compact and hemmed in by the water - having a car there is unnecessary and a hindrance (it will just sit parked). You can see (mostly) everything on foot - it's that small. There are buses that can bring you up to the town above, which is also interesting. A gondola takes you to the beach and Isola Bella below. Four nights there seems like a bit much (you can basically see the whole town in a full day) unless you plan on doing side trips, like Mt. Etna and/or you want a very leisurely time there. The main drag is pedestrian-only and it's not very long. There are buses that run to Catania airport from Taormina - just check the schedule. I would opt to stay the last night in Catania if you want a leisurely exit.

http://www.planetware.com/tourist-attractions-/taormina-i-si-ta.htm

You don't need a car in Palermo, therefore rent one only when you leave town.
I think 3 nights in Palermo are probably sufficient.
I don't think you need 4 nights in Agrigento either. The main sight (the Valley of the Temples) can be seen in a few hours. Nearby you can visit also the the peculiar cliff formation called Staircase of the Turks.
I think 2 nights in Agrigento would suffice. Maybe, after Palermo, you may want to spend 2 or 3 nights in Western Sicily in the Scopello area (quaint village) from where you can visit the Segesta Temple, the Zingaro Reserve, the hilltop village of Erice, and a few more things, including the many temples of Selinunte.
Four nights in Siracusa may be one too many, but I guess you can use them to visit Noto from there. However consider visiting Piazza Armerina and the Villa del Casale between Agrigento and Siracusa.
Taormina is a small town, that can be visited in less than a day. You don't need a car for the town itself, which is largely pedestrian only area, however, I would keep the car to visit places nearby, including Mt. Etna or others. Check with your hotel if they have parking arrangements, although Taormina has a parking structure sort of inside the mountain. That's where I usually parked (but I generally prefer to stay at the bottom of the hill in Giardini Naxos, because I like the enjoy the beach.

From Siracusa, you can easily visit Noto, Ragusa, and /or Modica (the other Baroque UNESCO sites)...also, I wouldn't miss the excellent Paolo Orsi Archeological Museum and the Archeological Park.

I would keep the car in Taormina to visit the area around it: maybe a drive around Mt. Etna. The area leading up to the mountain is beautiful. I agree with only a couple of days in Agrigento, but I would spend at least four days in Palermo. (no car needed while there) If you are worried about getting the car back to the airport in Catania, take it there the day before and spend the night in Catania.

I also loved Piazza Armerino to see the mosaics. We stayed at Farm Home Torre di Renda. Very reasonable, wonderful dinner, rooms--some better than others but the location was spectacular.

We stayed in Giardini Naxos by the beach and took the bus during the day to Taormino which avoided the parking situation. We stayed at B and B Miramare, I believe $100 per night with a great breakfast overlooking the beach.