Paphos has a reliable and inexpensive bus system connecting Kato Paphos and the upper town. Hiring a car is a good way to explore the surrounding areas but the main parts of town can be easily navigated on foot.
Municipal taxis are available and tend to be reasonably priced, servicing Paphos and its surrounds. But it's recommended that these are hailed rather than pre-booked as drivers tend to turn on meters when leaving to pick up passengers. If taxis are unmetered be sure to agree on a fare before setting out for your destination.
For the more adventurous, hiring a scooter can be an exciting way to explore Paphos but accident rates are very high. If you are brave enough to enjoy this mode of transport then be sure to invest in a helmet. In general, Paphos is not a difficult or expensive city to get around.