Sicily's capital is a multilayered, vigorous metropolis with a strong historical profile; approach it with an open mind. You're likely to encounter some frustrating instances of inefficiency and, depending on the season, stifling heat. If you have a car, park it in a garage as soon as you can, and don't take it out until you're ready to depart.
Palermo is easily explored on foot, but you may choose to spend a morning taking a bus tour to help you get oriented. The Quattro Canti, or Four Corners, is the hub that separates the four sections of the old city: La Kalsa (the old Arab section) to the southeast, Albergheria to the southwest, Capo to the northwest, and Vucciria to the northeast. Each of these is a tumult of activity during the day, though at night the narrow alleys empty out and are best avoided in favor of the more animated avenues of the new city north of Teatro Massimo. Sights to see by day are scattered along three major streets: Corso Vittorio Emanuele, Via Maqueda, and Via Roma. The tourist information office in Piazza Castelnuovo will give you a map and a valuable handout that lists opening and closing times, which sometimes change with the seasons.