We have rented cars in STT many times. If you are docking at Havensight (which I think DCL still does), there are several rental companies very close by. I have reserved online and then walked to the desk to pick up the car. Yes, the roads are curvy and mountainous and you do drive on the left side of the road. It can be treacherous. Parking in town can also be difficult. Having said all that, if you want to see the island renting a car is going to give you much more flexibility than a shore excursion or even a taxi. It's easy enough to do. Just be careful. And enjoy. Oh, and don't leave valuables in the car when you get out to go somewhere.
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