The best way for a tourist is perhaps still by taxi although Panama City has recently added new metro buses and a new metro line. Panama is not particularly pedestrian friendly, but you can stroll the Cinta Costera (Coastal Highway), Avenida Central and the Amador Causeway.
It's easy to rent a car in Panama City and allows for the greatest flexibility, but driving is another story. Major rentals include Thrifty, Hertz, Dollar, National, Sixt, Avis, Alamo and Budget. Stick to the speed limits since speed traps are common and have your license and passport available for police point checks. When renting a car, you will have to buy two obligatory insurances.
Popular biking routes include the Amador Causeway and the Cinta Costera, equipped with bike paths, and up Ancon Hill. The more adventurous may choose to cycle on Panama City streets - a death-defying experience in a city that is definitely not bike friendly except for Sundays from 6 a.m. to noon, when the city shuts down some streets to traffic to create the Ciclovia from Panama Viejo to the Cinta Costera. Bike rentals are available in several locations in Casco Viejo and on the Amador Causeway.
Inaugurated in 2014, Panama City's Metro currently links Los Andes to the Albrook Terminal through the city center, stopping at 12 stations. More lines are planned as part of rapid transit system remodel of the city. Use the Metro Card, available for purchase at points of sale throughout the city and also used for the city's bus system.
Metro Buses run all around the metropolitan area but make sure you study the routes beforehand (http://www.mibus.com.pa/rutas/). You'll have to buy a pre-paid Metro Card ($2 for the card + fare, cash only) to ride these white and orange buses. Tickets are available at point of sales throughout the city (check tarjetametrobus.com) and can be used on the new Panama Metro, which connects to the bus lines.
And before the new Metro Bus system, there were the Diablo Rojos (Red Devils), the colorful school buses that belched smoke and ignored traffic norms. You'll still see these in the streets, so if you are feeling adventurous, hop on for a quarter. They are also used as party buses around town.
You can find a yellow cab nearly anywhere, but the trick may sometimes be finding one that takes you where you want to go and at a fair price since cabs are unmetered. Expect to pay a bit more than locals, but make sure it is within reason and ask to see the mandatory fare card which will show you the regulated fares by zone. You may also find that drivers stop to pick up other passengers along the way. Uber has also set up shop and offers safe and reliable service.