Flights to Cyprus arrive into either Larnaca or Paphos in the south and Ercan in the north, however if you fly into Northern Cyprus it is not possible to travel to the South. If you are arriving into Cyprus on a package holiday, a transfer may well be included in the price, although taxis are widely available.
The best way of seeing the island of Cyprus is by hiring a car, and driving. Most sights can be driven to within a couple of hours. There are also some boat trips that run along the coast, and allow you to se the abandoned resort town of Famagusta, which has remained empty since the invasion of 1972.