The most convenient way to come to Famagusta is by air. Once you land in Ercan Airport, you can take a shuttle directly to Famagusta. The name of the shuttle company is KIBHAS. It takes almost an hour to reach Famagusta and it costs10 TL/person.
If you are travelling as a family/group, I wouldn't suggest to use Kibhas though. I have given the timetable below but they don't really stick well with the timetable. Plus, I have no idea how they fixed the timing because they never matched with my arrival or departure times. If you are travelling as a family/group, you can rent a car or just take a taxi. For the taxi, you should pay something around 50-60 TL (Make sure you fix the price before you start your journey).
In case you feel like using Kibhas to go from Ercan Airport to Famagusta, here is the timetable:
Departs Ercan: 09:30 - 12:00 - 16:30 - 19:30 - 23:45
Departs Famagusta: 05:00 - 09:00 - 13:30 - 16:30 - 18:30
The public transportation does not exist in Famagusta. You either walk or take a taxi or rent a car. The taxis generally fix the prices in Famagusta. It is better to talk with your hotel receptionist or the owner of any local store to learn the current taxi fare fixing for the city. It is not written anywhere but it is fixed. In August 2011 for any travel within Famagusta, the fare was fixed at 7 TL. When the taxi driver notices you are not a local, he can behave as if he is unaware of the fixing and may try to fully charge you. In such a case, feel free to pay 7 TL and leave the car.
If you don't have a rental car and you want to go to Nicosia or Kyrenia from Famagusta, you can take a bus. The buses are run by a company called "Itimat". They don't have fixed timetables so you should go to their office or call them before you plan your trip to Nicosia or Kyrenia.
You can easily go to the old town of Famagusta, as a day trip! Just go (by car) to the Agios Dometios crossing point, to cross over to the north part of Cyprus. Then follow the signs to "Gazimagusa" (Famagusta), which mainly involves travelling eastwards (just a long straight road) through the (dry) fields of Mesaoria. You will reach the old town of Famagusta in 30-40 minutes. The old town is beautiful, with lots to see. However, the "modern" part of Varosha, is off limits...
You can take dolmush between Girne(Kyrenia) and Gazimagusa (Famagusta). Locals say "Kombos" to dolmush in Northern Cyprus.It's the cheapest way of travelling around Northern Cyprus.
PS: A Turkish dolmush (DOHL-moosh) is a jitney, a shared taxi or minibus running a pre-determined route, with each passenger paying only a portion of the normal fare. Its very common in Turkey and Northern Cyprus.
Although there is a shuttle service between Ercan Airport and Famagusta, taking a taxi right away is more convenient (in case you don't want to rent a car directly). It roughly takes an hour to reach Famagusta and be sure that the taxi is overcharging you if you are asked to pay more than 60 TL. To avoid this, the best way is to fix the price with the taxi driver before you get in the taxi and start your journey.
Getting to Famagusta from Girne (Kyrenia) by public transportation is really east. There are shared minibuses - called dolmus - leaving from Girne every hour (on the hour) from about 8 AM until 7 PM. From Famagusta to Girne they leave every hour (on the half hour) until 6.30 PM. There are set stopping points, one of them (in Famagusta) being near the huge Ataturk statue in the new city, but it's best to ask the tourist information just to be on the safe side (things change...)
From Famagusta it's easy to get to Nicosia, too (again by dolmus) or else to the Greek part of Cyprus. There are a couple of buses a day heading to Aya Napa on weekdays, otherwise you can take a taxi to the border (it'll cost about 10 Turkish lira, the journey is about 10 minutes) and then ask a border official to call you a taxi in Aya Napa, which is only 20 minutes away. it shoulldn't cost more than 20 Cypiot pounds.
Not all guidebooks are up to date - but this border crossing is open. Some, including the LP, still claim it's closed.