Easyjet, Ryanair and Jet2 all fly to Paphos and Larnaca on a daily basis from all main London airports.
If you are staying in Paphos, Limassol or Ayia Napa, taxis from the airport are fairly cheap. If you are heading north having landed in the south, you will need to get a bus or taxi to cross the border.
Otherwise, to land at Ercan airport in Northern Cyprus, all flights go via Turkey where you can have to wait on the ground for a minimum of 30 minutes.
Check TNT Flight Search for flights to Cyprus.
The best way to travel is to hire a car. They are cheap to hire and it gives you the freedom to explore whenever you want.
There are dozens of car hire firms at the airport and all are competitively-priced. Public transport can be patchy at best.
There are no trains, but local bus services are fine to flag down and use to explore, and most larger towns (like Paphos) have hop on/hop off bus services. Make sure you check the timetable though, as services can be infrequent.
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