Panama City's traffic is bad — seriously bad at rush hour — but the city is manageable once you understand the geography. The key areas for visitors are the Casco Viejo (far western tip of the bay), the modern downtown and Marbella/El Cangrejo hotel districts (central), Miraflores Locks (northwest, 10km), Panamá Viejo ruins (eastern edge, 6km), and the Amador Causeway (just southwest of the Casco). None of these are walking distance from each other.
The Panama Metro has two lines and is cheap, clean, and efficient for the corridors it covers. Line 1 runs east–west through central Panama City connecting Albrook (the main bus terminal) to the eastern residential areas, passing through Cinco de Mayo near the Casco and several central stations. It does not serve the Casco Viejo directly; the nearest station (5 de Mayo) is about 20 minutes' walk. Line 2 extends east toward the new city. The Metro is worth using for longer cross-city trips.
Taxis are ubiquitous, inexpensive, and the default for most visitors. As a rule, agree on the price before getting in, or use Uber, which removes all price negotiation and is widely available. Uber is generally a few dollars more than a taxi but operates predictably and tracks your ride.
The Casco Viejo itself is entirely walkable — most of the relevant streets are within a 10-minute walk of each other. The Amador Causeway is good for cycling; bicycle rental is available near the causeway entrance. Traffic is worst 7–9am and 5–7pm on weekdays; if you are visiting the Canal or the ruins, aim to arrive before 10am or after 3pm.