Cyprus is small enough that most of the island is reachable on a day trip from Ayia Napa. Buses connect the major cities, and rental cars and scooters are readily available for more flexibility.
The sea caves east of town, about 5 km along the coast toward Cape Greco, are the easiest outing -- you can cycle or drive there and spend a few hours exploring the rock formations and swimming. Further afield, Larnaca is about an hour west by bus and has a good waterfront promenade and open-air market. Jeep safaris run north into the occupied territory, passing through Famagusta and up to Kyrenia, where the harbour and Bellapais Abbey are worth the trip for anyone interested in history.