The only direct flight from the UK is Ryanair from Stansted. There are scheduled flights from Heathrow, changing at Milan or Rome - see Expedia.co.uk. To avoid having to trek down south you could always fly from Leeds-Bradford to Bergamo or Rome with Jet2, and then fly from there to Palermo.
The airport is a good 20 miles from Palermo. A shuttle train operates between the airport and the Central Station every hour between 5.40 and 21.40, with the odd additional service in between (full timetable). Journey time is approximately 55 minutes, and tickets cost €4.50. There are a few intermediate stops which could suit you better than the Central Station depending on where your hotel is. There's also a bus (no 29) every half an hour; it costs €5,and you buy your ticket on board. A taxi will set you back around €50.
Guides & info
Palermo tourism site (good site)
Rough Guides (select Italy then Palermo)
SideStep (American site, but good)
Airport rail service (unofficial site, in English)
Rather than staying in Palermo itself (which let's face it has a bit of a dodgy reputation, particularly since West Ham's visit), you might instead prefer the seaside resort of Mondello. It's about 5 miles north, but there's a good bus service from Palermo and it looks really nice (click here for some pics). A good value hotel is the Addaura Hotel Residence Congressi, which can be booked via the Venere.com link below. There's a bus from outside the door (no 603) direct to the ground!
Venere.com (enter a discount code of BP93PTS on the Booking Details screen to get 4% off)
Palermo tourism list
The bus service is operated by AMAT. There's a flat fare of €1.00 for any number of journeys made within ninety minutes (including out as far as Mondello), or you can buy an all-day ticket for €3.50. Tickets for the minibus service around the historical part of the city cost €0.52 for a day. All tickets are available from the AMAT glass booths outside Stazione Centrale, or at the southern end of Viale della Libertà, in tabacchi (tobacconists - look out for the big T on the sign outside), or wherever else you see the AMAT sign. Validate tickets in the machine at the back of the bus the first time you use them. People getting caught without tickets by inspectors is apparently a major source of income for the bus company; you've been warned!
Interactive bus route map
Club and ground
Palermo play at the Stadio Renzo Barbera. The ground is a couple of miles out of the centre (map). You catch the number 101 bus.
Palermo official site (in Italian, but with a link to a cut-down version in English)
www.stadiumguide.com (some good pictures of the ground)
Stadionwelt.de (German site, but with loads of good pictures of the ground)