Loads of places to stay in the old town. Ponticello apartments are very very nice. Don't stay too close to the train station, gets noisier and dirtier past via Divisi. Area just north of Piazza Politeama is also really nice to stay in and has some good restaurants.
Trattoria al Sorriso by the station is superb. Great food at pretty low prices. Have fish obviously. Trattoria Trapani is a proper old school cheap trattoria, also next to the station. Pizza e Cucina on via 20th Settembre is my old haunt. Pretty unassuming but just superb pizza and cold beer. Vecchio Club Rosanero is a Palermo FC themed trattoria, pretty decent cheap grub. Ballaro of all the markets probably has the best places to eat, though even that one has become a bit touristy - Kemonia is good if memory serves. Trattoria Bersaglieri is where we always used to go on Saturdays. Bit dodgy and mafiosi but huge portions for very little cost.
There are loads of places to eat. You will eat well there. I'll wrack my brain for more.