Do I have to fly into Panama City when coming from another country, or are there other airports that receive international flights?

While the Tocumen Airport in Panama City is definitely the main hub for international flights (and probably for cheaper flights as well), there are two airports in Panama receiving international flights, mainly via Costa Rica: Bocas del Toro and the Enrique Malek airport in David.  The David airport has been undergoing major expansions, and there is an active campaign to make it ready for international flights from the U.S.  in the coming few years.