Catacombe dei Cappuccini- A remarkably unique burial catacomb where the humans are preserved in their clothes. It is bizarre and wonderful to see, and is the largest collection of mummies in the world.
Santa Maria dello Spasimo- An unfinished Catholic church that is a peaceful place to take photos, and relax. A church without a roof.
Piazza Pretoria- A gorgeous square with a large impressive fountain, Fontana Pretoria, nicknamed the square and fountain of shame. My favorite part of the fountain is it's animal sculptures, including a rhinoceros, which is rare to see as a fountain sculpture.
Mercato il Capo- This where I went to get arancini, a small fried rice ball filled with meat and peas. I had mine at Friggitoria Gastronomia da Arianna, and it was yummy.
I Sapori del Mare- I got the pasta with ricci di mare, and it was great, but he real show stopper was the swordfish. It was exquisite. The service was terrific and the atmosphere was a traditional spot in Palermo.
Pani Ca Muesa Porta Carbone- I came here to try milza (spleen), a typical street-food item of Palermo. It was my first time eating a spleen sandwich, but I think it was pretty good. Adding the lemon and cheese opens up the flavors.
Bar Garibaldi- A cozy bar that plays good music, music you haven't heard in awhile like Portishead, and the bar is filled with locals.
Caffe del Teatro Massimo- This is a perfect cafe. It's beautiful and calm, the service is efficient, and you might not ever want to leave. Before you go to the cafe, you can take a look at the steps of the theater that were in Godfather Part III (you know the good one :)) or you could tour the entire theater. It is gorgeous.
Botteghe Colletti- A bar where you could hang out for hours. Cool music and relaxed atmosphere. Mixed groups of locals and tourists.
Cioccolateria Lorenzo- A lovely cafe to stop off at for a mid-day break, and indulge in some delectable sweets. The staff is friendly and you have many different coffee drinks to choose from. I ordered a latte and it was tasty.
Bocum Mixology- A pleasant place to try a delicious cocktail in a comfortable environment. The name is a little lame, yes, but don't let that stop you from going. The bar is cool.
I recommend renting an apartment in Palermo. I don't think the hotels are really up to snuff, and it's a better way to get the local experience.