The Republic has airports at Larnaka and Pafos (spelled 'Larnaca' and 'Paphos' on many airline schedules), with flights from most of Europe and the Middle East. North Cyprus has an international airport at Ercan, but only Turkish airlines flies there. Passenger ferry service to the Republic was suspended in 2001; check the Salamis Lines site for updates. There are ferries from North Cyprus to Turkey, but you cannot exit Cyprus this way unless you entered from Turkey.
Bus services run within and between towns every day except Sunday. Those between major southern cities are cheap, frequent and efficient. If you want to travel on a Sunday, you'll need to get a service taxi. They're good for most places in the Republic, but they tend to take roundabout routes. North Cyprus has minibuses between Kyrenia and North Nicosia. You can also use service taxis to cross between the South and the North. It's easy to hire cars and bikes all over the country.
If you enter Cyprus from Turkey, customs officials from the South are authorised to fine you for entering through an illegal port. In practice, this policy is not enforced.