The canal is the obvious one and genuinely worth the trip, but the city also holds the ruins of its own first incarnation, a rainforest inside the city limits, and a 500-year-old grid you can walk in an afternoon.
Miraflores Visitor Center, Panama Canal
Viewing terraces over the Miraflores lock chamber, open 08:00–18:00 daily with ticketing until 17:00. Non-resident tickets run $7.22–$17.22 depending on which levels and the film you include; check the day's transit schedule on their site so you arrive when a ship is actually locking through.
Parque Natural Metropolitano
About 232 hectares of tropical dry and humid forest inside the city at Ancón, with nine marked trails and viewpoints over the skyline. Trails are open 07:00–16:30 every day including holidays; foreign visitors pay $5 adult and $3 for children 3–17.
Panamá Viejo
The ruins of the original Spanish city founded in 1519 and destroyed in 1671, now a UNESCO-listed archaeological site with a museum on the eastern edge of the modern city. Run by the Patronato Panamá Viejo — check current hours on their site, and go in the morning, as there is very little shade.
Casco Viejo on foot
The walled city built in 1673 to replace the one that burned, and now a grid of restored plazas next to blocks still mid-restoration. Walking it costs nothing and takes an afternoon; the churches, the seafront Paseo Esteban Huertas and the plazas are the spine of it.
Biomuseo, Amador Causeway
Frank Gehry's brightly panelled natural history museum on the causeway out into the bay, built around a single subject: how the isthmus rose and split one ocean into two, remaking the wildlife of two continents. Their website's security certificate is currently broken, so check hours by phone or at the door.