Soberanía National Park and Pipeline Road (30 minutes) — The Pipeline Road is the easiest and most rewarding day trip from Panama City; see the Pipeline Road listing for detail. Hire a guide, leave early, and you will be back in the city by midday.
El Valle de Antón (3 hours west) — A village in the crater of an extinct volcano, with a cooler climate, good walking trails to waterfalls and the famous square trees, a Sunday handicraft market, and a small orchid nursery. A popular weekend escape for Panama City residents; overnight stays are possible if you want to do more walking.
Colón (1 hour north) — The city at the Atlantic end of the canal is rough around the edges and requires care, but the contrast with Panama City is dramatic: Caribbean architecture, a different pace, a different population. The Gatún Locks on this side of the canal are also worth seeing, and the town of Portobelo (a further hour east) has significant Spanish colonial fort ruins. Go with a guide or at minimum with local knowledge.
The Pearl Islands (ferry or short flight) — An archipelago of 90-odd islands in the Pacific, 90 minutes by ferry or 20 minutes by light aircraft. Isla Contadora is the main tourist island: snorkelling, whale watching (humpbacks July–October), and the beach lifestyle that Panama City doesn't provide. The island was where the Shah of Iran lived in exile briefly in 1979 and where the first season of Survivor was filmed. Day trips are possible; an overnight gives you more.