The airport is located 20-25 kilometers to the East of Panama City, depending where your hotel is. There are several options for transport, Buses for the brave with backpacker type luggage, Taxis in a variety of condition, from almost new to total wrecks, or a prebooked pickup.
Buses: NOT recommended unless you are a really adventurous type travelling light, and definately not after dark. City buses here tend to be overloaded so you will probably be standing, and used only by the poorest folks, so keep your eyes open for opportunists. Try to stay near the front entrance of the bus, the emergency exits are often welded up to stop folks slipping out and avoiding the fare, and you don't want to be trapped in the event of a mishap.
You will need to walk out of the airport to the main road, and wait for an ancient school bus painted in crazy colours with little maintenance. The routes are marked on the front windscreen in fancy script. A bus to 'Via Espana' ($1 last time I was crazy enough) will drop you in the centre of the city where you can get a cab for $1/$1.25 to most hotels.
Taxis: When you come out of the airport, chances are a 'despatcher' will buttonhole you and offer to get you a cab. The 'official' rate to the city is $25 for a cab for 1 or 2 people with normal luggage. However, many drivers will see 'Gringo' and rates of $50 are not uncommon if they think you are naive enough. Check the fare with the driver or despatcher before you accept. Also take a look at the cab, don't just accept the first one offered if it looks in bad condition, if you are paying full rate tell the despatcher to get you a decent ride.
A technique I have used to get a discount is to refuse the despatchers help, and phone/pretend to phone your own driver. The despatcher will ask who you are calling and I just say I'm calling my own cab, because its only $15. You won't get the most modern cabs this way, but, unless its very busy, the despatcher will have a number of drivers who will take the fare and will fix you up.
Prebooked: The safest way. You know you have a decent car waiting for you, an english/spanish speaking driver, help with your luggage and a guide. Same price as the official taxi rate, $25. Email
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Car Rental: Is available at the airport, but more expensive than in the city unless you prebook over the Internet. Check the car over very carefully and get any damage listed with 2 copies, one for them and one for you, both on headed paper and signed. Rental companies here are notorious for adding 'damage' when you return the car. Also check the amount for insurance, insurance cover is expensive here due to the number of 'fender benders', your credit card company may offer an acceptable alternative.
If its your first trip here, I would recommend using transport into Panama city, and assess the traffic situation before commiting to a rental car. Driving here is not for the faint hearted, the city is close to gridlock at times, there's no logic to the junction layouts, many signs are missing and others often get ignored. Unless you have a specific need for a car, I would avoid the stress and use taxis which are plentiful and cheap.