There are two airports in Cyprus, the main one being Larnaca, which can be reached from a fairly wide range of destinations across the UK, and Paphos, which is gaining popularity and therefore there are more cheaper flights available.
Package holidays normally provide transfers between the airport and your hotel, otherwise there are shuttle buses to Limassol and Nicosia. You can also hire a car which will give you freedom to explore the island during your stay.
Cyprus has a fairly good bus network that covers all the major destinations both within and between towns, and taxis are very prevalent and not too expensive for shorter distances.