It takes about 50 minutes to drive the island from end to end from Vlychada to Oia.
Santorini has a public bus service, with tickets costing from 1.60€ to 2.20€. There are no daily or weekly passes available. Buses run between every 30 minutes to every other hour. Timetables are available at website ktel-santorini.gr. Buses are air-conditioned but may be overcrowded during the high season. If you prefer getting a private or shuttle transfer from airport or port to your destination or even a private tours service throughout the island, there are companies such as Premier Santorini offering such services.
You can rent a car from 25€ per day, an international driving license is recommended. Without an international license you can still rent a car, however there might occur insurance problems in case of an accident. Since the island is quite crowded with most drivers being tourists, accidents happen quite often. Scooters and 4-wheelers are available from 15€ and 30€, respectively.
An ATV is probably the most popular method to get around, however not the safest one. It is quite cheap, and Santorini is small enough to explore with an ATV. You will need your local driver’s license to rent one, just make sure to find a trusted vendor (ask your hotel).
Taxi is always an option, however, make sure to book one in advance since there are only 30 taxis in the island and the availability is limited during high season.
Santorini is not considered bike-friendly. The island is quite hilly and there is a lack of infrastructure, there are no dedicated bicycle routes, so you must share roads with cars, buses, etc.
In the last 2-3 years some guided bike tours made their appearance, so it would be the best option to find one.
The best thing to do in Santorini is to explore every corner on foot.
Hiking from Fira to Oia is one of the most amazing activities in Santorini. The entire route is about 10km (6 miles) and takes 3 to 5 hours, however it is considered quite easy. The path is a mix of paved pedestrian streets and hiking trails.