Public transportation is widely available and stops at or near most destinations of note. It is a good idea to buy a transportation pass, which you can purchase for 1-7 days. A 1 day pass is $15 per person and is good for both buses and ferries, across all transportation zones. See here for more information. Remember that most locals commute into Hamilton to work, so the buses and ferries are quite busy during morning and afternoon "rush hours."
Buses
The central bus terminal is located on Washington Street, near City Hall and is the hub for the 11 bus routes across Bermuda. Schedules are available at at the terminal wherever you purchase tokens or a transportation pass. Take the times on the schedule as a guideline, not a rule. If you need to be back to the port at a certain time, don't cut it too close. Hamilton is about an hour bus ride from either St George's at one end to the Dockyard at the other.
If you want to catch a bus, look for the blue and pink poles along the side of the road. Pink poles are for buses heading to Hamilton. Blue poles are for buses heading away from Hamilton.
Water Ferries
Sea Express is the company which operates the water ferries and they can be a quick way to get from one point of the island to another. Plus, its a nice way to see Bermuda from a different vantage point. There are four different ferry routes, of different colors and they follow the schedule fairly closely.
Taxis
Taxis are available at the ports, St. George's and outside of most major hotels and resorts. Fares are regulated by the government at a current cost of $6.40 US for the first mile and $2.25 each additional mile for 1-4 passengers. Taxis do offer tours of the island with prices that start around $40 per person for an hour. Because the speed limit is only 22 mph (the roads can be narrow with lots of twists and turns) you wouldn't get too far in just an hour. Usually there is a 3 hour tour minimum. Look for taxis with a blue flag - these have been vetted as qualified tour guides.
Scooters
You will notice that scooters are a main mode of transportation for locals. While rentals are available around the island, most cruise lines don't recommend their passengers doing this. It's an economical way to tour the island, however there can be some crazy drivers and the roads in some spots are so narrow that two vehicles can barely pass each other, let alone a scooter on the side of the road. Plus, people drive on the left side of the road.
Car Rental
Only locals can own or rent cars, primarily because driving can be hazardous, with blind curves and narrow roads, most of which have no shoulders.