The country of Panama as a whole enjoys a muted destination profile compared to its neighbor Costa Rica, and not many people take the time to explore it. This does not mean that there is nothing to explore, especially when it comes to the capital city.
The historic town center, Casco Viejo ♥♥, which is part of a UNESCO WH inscription, is in places very picturesque, a mishmash of old colonial styles, and quite literally a town of thousand balconies. There are some ruins and building carcasses right in the middle of renovated blocks, which creates an interesting juxtaposition. Several outstanding churches are definitely worth stopping by: La Merced ♥♥ with its thin wooden columns and varied chapel altarpieces; San Jose ♥♥ with the Golden Altar (and if you come around Christmastime, with an altogether incredible Nativity display); the Metropolitan Cathedral ♥♥ with its outstanding facade of weathered stone flanked by gleaming-white towers and the colorful marble altar; as well as San Francisco ♥ with its relatively modern-looking altar and chapels.
Nighttime Casco Viejo can be enchanting in places, but also somewhat deserted in others. Festive lighting was frequent during Christmastime, but I have no evidence of whether it remains on at other times.
The second part of the same WH property is Panama Viejo ♥, the site of the original settlement in the area that was abandoned in the 17th century. It is located to the northeast of the city center and is practically all ruins, with just a couple of standing structures, most notably the cathedral tower. The museum on premises offers a very detailed picture of the history of the settlement.
Visiting the Panama Canal may be the only true unmissable sight when in Panama City. The Miraflores Lock ♥ provides a great overview of how the canal works and allows reasonably close observation with ongoing commentary flowing from the speakers. Note that things proceed very slowly and the place is usually fairly crowded.
There are a couple of museums in Casco Viejo that may be interesting to visit given extra time, such as Museo de La Historia de Panamá or the Panama Canal Museum. A striking building on the Amador Causeway is Biomuseum, and further on Naos Island is the Punta Culebra Nature Center. One other further afield attraction that may be worth consideration is the Gamboa Rainforest Reserve, with several places of interest including Monkey Islands.
As you walk through town, you may decide to taste raspa’o, the local concoction of shaved ice fortified with condensed milk and fruit flavors. Refreshing, although in my view it is below Sicilian granita in terms of enjoyment.