Larnaca International airport is the main entry point for visitors to Famagusta. The airport is about 7km from Larnaca city and takes most flights in and out of Cyprus. There are regular bus services (No 22 and No 24) from the airport to the town centre from 6.20am to 7pm and taxis also lie in wait.
You cannot cross directly into Famagusta from south Cyprus. Instead, you must cross the Green Line into the Turkish-occupied territory. The nearest place to do this is from the Strovilia, crossing near Agios Nikolaos, which is found inside the British sovereign military base area. The other is through Nicosia.
In north Cyprus it is best to travel by car. Famagusta has no public transport and there is no regularly scheduled public transport across the Green Line although private tour operators do operate in the peak tourist season.
Ferry boats also run three times a week from Famagusta to Mersin in Turkey and there are also some ferry boats that go to Tasucu. The ferry to Tasucu however, takes about eight hours so it is better to travel to Kyrenia to catch a boat as the journey time from there is less than three hours.
Driving is the way most people get around in Famagusta with a rental car. There are no cycle rental firms that I could find. It's possible to travel on foot, but the streets and paths are in very poor condition and you need to take care not you accidentally cross into areas held by the Turkish army or the UN. If you do you can get arrested. Taxis are probably the safest option for independent travellers. These are widely available and inexpensive for trips or tours.