As others have stated, I would recommend renting a car, but this depends upon where you are staying and what you are doing. It is cheaper if you rent a car from anywhere that is not at the airport. Be sure you schedule your rental in advance or you will most likely be out of luck when you get here.
We drive on the left side of the road... or the center... very seldom on the right. The roads on St. Thomas are narrow and hilly and the locals drive a bit crazy. You won't see cars without a few scrapes or dings, typically. I see accidents from time to time, but have never seen anything too bad. There really isn't a road on St. Thomas that you get get above 40 MPH for more than a few seconds.
An arm out the window is usually telling you to stop, not to pass. The law states that you are allowed to stop on the road "for a while" to talk, lime about, or whatever. This is one of the more annoying things that can happen to you... if you get road rage easily, let somebody else drive. Remember, don't plan on getting anywhere quickly... island life is slow, me son... if you try to speed it up, you will regret the wasted effort.
It is not uncommon to see accidents around the airport. These are usually tourists that forget which side of the road to drive on.
As others have stated, the horn is used to state "thank you" by a quick beep or two. A long blast is considered a rude, get outa the way and probably won't get you anywhere. When somebody stops to let you in traffic, you should reply with a quick beep. They will usually answer back the same way.
The drives, except for most taxis, are very polite and will tend to stop and let you out. You have to be a bit aggresive to let other drives know that if they do stop, you are going to butt your way in and get out their way quick. If you sit back, you will be waiting... forever.
All taxis are set price and easy to identify by the licence plate that reads "TAXI" on it. Don't take rides from anybody that doesn't have a taxi licence plate. The prices are fixed by the legislature and the taxi association.
You can also use the gypsy taxis... they drive all over the island. Stick out your index finger as one comes down the road to ask him to stop. Hope on board and take a tour. They will announce where they are going, downtown, redhook, havensight, etc as they loop around the island. When you are ready to get off, simply press the little doorbell button and the driver will stop. Five dollars a head, a bit slow and indirect, but cheapest by far. (Locals ride cheaper... but the drivers get upset if tourists try to only pay a buck or two...)
Gas here is not cheap. Right now we are paying around $2.80 (or more) a gallon. If you drive, be prepared to budget a bit more than you are used to paying in the states.
I would avoid those scooters like the plague. I have come closer to running over more tourists on scooters than all other driving mistakes combined. As a local driver, the scooters are extremly annoying.
(Hey Chris, how is state side life? Your signature is on my PADI Open Water Certification... I was in the last class you taught down here with CSDC before you moved back to the states. Small world...
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