Ayia Napa is compact enough to walk around, and most hotels, beaches, and nightlife are within a 15-minute walk of The Square. For beaches further out (Konnos Bay, Ayia Thekla), you will need transport.
Local buses connect Ayia Napa to Protaras and Paralimni regularly during the day. Scooter and quad bike rentals are popular -- just make sure to check the brakes and condition of any vehicle before you ride. Taxis are available but can be expensive for short trips; agree on a price before getting in.