The first thing that visitors to Bermuda should know is that rental cars are NOT available.  In lieu of this, there are really five feasible options for travelers that want to get around the island:

Taxi
The easiest, but most expensive, option if you're traveling a lot, taxis can still be the best way to see the island.  In fact, many drivers are more than happy to schedule an hour to two hour long tour of the island at a fixed rate of $50.00 / hour. 

Please note that all taxi fares increase by 25% on Sundays and Public Holidays.

Gratuities are not required but are expected.  

The fare to some hotels could run as high as $50 from the airport, so it's worth doing some research prior to your trip on transportation options.  The standard rate is $4.80 for the taxi, $1.68 for each additional mile with $1 extra per piece of luggage, Sundays, and public holidays.

Scooter
For the brave and fearless who want to get a taste of the local life, it's quite easy to rent a scooter to zip around the island.  All scooters rented to tourists have a maximum speed setting, but unless you're familiar with riding one, it's unlikely a new visitor would hit top speed on the first visit.  Mildly terrifying at first on the small winding roads of Bermuda, scooters are a blast once you get the hang of it.  Scooters are the best way to really experience the island if you are calm, confident and take your time. JUST REMEMBER TO DRIVE ON THE LEFT-HAND SIDE!  Long and laborious vs fun and invigorating on the scooter.  Drive with caution and it's a piece of cake.  Enjoy cruising South Shore Road with a view of the vistas.   Speed limit on the entire island is a maximum of 35 kmh (22 mph) and riding on the left side is much easier in a scooter than in a car, and just like home if you are from one of the 50 or so countries in the world where this is the norm, including India, Australia, UK, South Africa etc.  If you're driving along and the locals cars are lining up behind you (single lanes roads everywhere), just pull off into one of the many bus stops and let them pass you by. 

There are only three main roads in Bermuda, i.e. North Shore, South Shore and Middle Road and all go from East to West. Generally speaking, if you want to see the island, then find one of the three - and you will get there. 

Most agencies will either pick you up and take you to their nearest store or deliver and pick up the scooter directly to your hotel for no additional charge.  They will give you a lesson on how to drive it and will ensure you feel comfortable on it before you take off on it.  The main rentals are Elbow Beach Cycles, Eve's Cycles, Smatt's Cycle Livery, Oleander Cycles and Wheels Cycles.   Helmets are included (and required) and can be stored and locked under the seat.  There is a wire basket on the back of the bike to hold your goods, but be mindful that theft happens in Bermuda as anywhere else in the world, so don't put valuables in the basket.  In Nov 2006 Wheels appeared to be the cheapest (saved $50 on a 5 day rental).

Caution:  Riding scooters in Bermuda can be dangerous.  If it is raining the roads can become very slippery.  Many tourists get hurt while riding scooters either by burning themselves or by road rash.  Be careful when riding with a passenger.  Take the bus or a ferry. (Response by rosethescimom)

Google Maps is very poorly updated in the area.  Do not expect good results.  Locals are very friendly and happy to point out (proper) directions (2013)

 

Ferries
From the major cities (Hamilton, St. George, Dockyard), Bermuda offers a ferry service that is quite convenient and the best way to travel if the times match up.   You can take a scooter on the ferry from the Dockyard to Hamilton and then ride back to your hotel from there.  The ferry from Dockyard to St George is seasonal (last ferry was Nov 24) and does not allow scooters.  They have their own website giving routes, fares and timetables.  Search under seaexpress bermuda 

Bus
For a public transportation option, the Berumda bus system is very clean, reliable and air conditioned in summer - a good option for those that want to save some money on transportation, or that want more flexibility than the ferry offers. The buses stop often so a ride can take a long time.  You must have exact change which is currently $3 for shorter distances. The most cost effective way to travel by bus, is to buy a sheet of tickets from the Bus Station. Cost is currently around $20. Also leaving town between 3 and 5 PM weekdays is rather challenging. There are two types of bus stop used; PINK and BLUE. If you get on any bus at a Pink stop - it will take you into Hamilton. A Blue stop will take you AWAY from Hamilton. They are request stops - so stick your hand out!

Tip: The bus drivers are very friendly.  Just tell them where you want to get off when you get on and they will let you know when to get off.  Be polite - say please and thank-you.  Most Bermudians expect you to greet them with a Good Morning or Good Afternoon.  It is custom in Bermuda to give up your seat for the elderly and for locals carrying groceries.  There are direct routes for the beaches and on Sunday they run special buses just for the tourists going to the beaches and to the cruise ships.  Ask at the bus terminal in Hamilton for your best routes to avoid numerous resident stops. Also, the bus drivers can refuse you if your bathing suit is soaking wet and you must have proper attire to board the bus.  Bring a change of clothes or a dry towel to sit on the bus.  The buses are very clean and have a fabric seat.(Response by rosethescimom)

Schedules and Information

Go Explore Bermuda provides access to local events, discounts and general information such as bus and ferry schedules.  www.goexplorebermuda.com

 Minibus

Another public transportation option is the Minibus. These vehicles are licensed by the Bermuda Government. They are designed to move groups but can legally transport any number of people throughout the Island, including to/from the LF Wade International Airport. Minibuses usually have a minimum capacity of 9 passengers up to a current maximum of 28 passengers (this maximum is determined by the seating capacity of the vehicle not by regulations). On average, most can carry up to 13 passengers. Excluding the public buses and ferries, this is the most affordable means of transportation to most places on the Island. The minibuses, like the taxis, will provide direct transportation to your destination. Fares are on a per head basis. Transportation from Kings Wharf to Horseshoe Bay Beach is $ 7 per person each way and tour rates are $ 10 per person, per hour - seven days a week. Drivers are very courteous. Gratuities are not required but are appreciated.