From Palermo airport to the city center it is possible to take a shuttle bus or taxi. The shuttle bus leaves every 30 minutes from outside the arrivals building (follow the signs airport bus-"Prestia e Comandé"). There are several bus-stops in town. The last one is at central station.
Taxi from Palermo airport to city center hotel cost 35-40 € depending on the location of the hotel. Better to check with the driver before.
It is also possible to fly to/from Trapani airport. Trapani airport is a Ryanair hub and it is not too distant from Palermo city center (about 1 hour 15 minutes drive). There is a bus directly connecting Trapani airport to Palermo city senter, see Terravision website.
Wish to rent a car to explore Palermo with its surroundings and the whole Sicilian island? Look at this website: http://www.autoeuropa.it/newae/booking/en/
Wish to rent a motorbike to explore Palermo with its surroundings and the whole Sicilian island? Look at this website: http://www.sicilymotorent.it
For those who have rented a car/motorbike and stay in a hotel without parking, a recommended garage with video surveillance is underground "Piazza Vittorio Emanuele Orlando" which is the square in front of the Court of Palermo so it is a very safe area.
With Google maps you can check the path to follow and the distance, by car or on foot from your hotel. You may drop off your luggage at the hotel and bring the car to the garage.
Rates are per hour but with a ceiling of € 8 for a maximum duration of 24 hour stopover.