Panama City is a great place. First, make sure you go to see the
Canal at Miraflores Locks. Via Espana (a road) offers some great and interesting shopping and Sol de la India (a store) near Hotel El Panama is a good cheap place to buy Panamanian souvenirs. Punta Paitilla and Avenida Balboa is an interesting area to explore metropolitan Panama City. Be sure to stay out of Casco Viejo (unless you take a cab right to the French Restaurant at the end of it), Chorillo (where most of the US bombing was concentrated), and be careful around the ruins of Panama Viejo as these are bad neighborhoods (the first two are downright dangerous but you'll figure that out the moment you see them). There are plenty of other attractions promoted, and if you want to go to the "interior" of the country, the area around David (Western Panama) is beautiful around the volcano (Boquete and Volcan) and so is Bocas del Toro on the North Coast close to the Costa Rican border. David can be an all-day bus trip (or commuter plane flight from Marcos Gelabert airport), and Bocas del Toro is realistically only accessable by commuter also. The Tocumen Airport Cab (white cars) gang will rape you with exorbant taxi fares (but all other cabs are prohibited from picking people up at the terminal) but downtown taxis charge 1 or 2 dollars to go within town (about $5 to Miraflores locks) and you can negotiate a cab to the airport from downtown for no more than $10.
Enjoy your time in Panama.