I wouldn't rent a car for such a short time unless I planned to stop in Monreale outside Palermo before dropping off the car. Two nights (about 1-1/2 days) doesn't provide enough time to see a great deal of Palermo, so I wouldn't want to divert any of those precious hours to other stops along the way. Monreale, though, is arguably the No. 1 sight on the entire island, so if a car helps you include that in your itinerary, I'm all for it. Don't try to drive in Palermo itself, though. The traffic is extremely challenging; there aren't as many traffic signals as we're used to. Crossing the street for me was sometimes scary as a pedestrian; I certainly wouldn't want to drive there. ViaMichelin estimates the driving time straight from Taormina to Palermo at 3 hr. 3 min. if you take the fastest route.
There is train service from the Taormina-Giardini train station (right below Taormina) to Palermo. Most trains take between 4 and 5 hours, with a transfer in Messina.
Via bus, I think you would have to travel first to Catania (Etna Trasporti does that run) (Etna Trasporti does that run) and switch to an SAIS Autlinee bus to Palermo. That would be a bit slower than the train.