Palermo to Taormina Transportation

Trains take 4-5 hours with one connection at either Messina or Catania and cost 15-20 euros. You might need a taxi from Taormina-Giardini station up the hill to your accommodation. I don't think there's a comparable bus alternative.

It can't be cheaper than a bus (although that depends on the number of passengers), although it may be faster. Buses are really inexpensive - the bus station in Taormina is no more than 5-10 minutes from the town gate. You would have to probably go to Catania first (this is easy by bus) and then pick up a direct bus from there. I know they run everywhere from the airport (which is the stop after the city center). Catania is a better hub for all sorts of long-distance trips. I would take a bus over the train - the train would likely take longer and have its own connection. Buses are very straightforward and there's very little traffic on the roads, at least when I went in September.

Here's the bus link:
http://www.interbus.it/Home.aspx

If you are only going to Palermo and Taormina, you don't want a car. If you are seeing other places in Sicily, a car is very handy.

Exactly. Parking will be a pain once in Taormina and expensive.