You might get by with taxis, especially if you are staying in Road Town, but if you are in a villa or an isolated resort, a car is a necessity here. Remember that driving is on the left even though cars are generally American-made. Tortola's main roads are well paved for the most part, but there are exceptionally steep hills and sharp curves; driving demands your complete attention. A main road circles the island, and several roads cross it, mostly through mountainous terrain.
Road Town's traffic and parking can be horrific, so avoid driving along Waterfront Drive during morning and afternoon rush hours. It's longer—but often quicker—to drive through the hills above Road Town. (And the views are great as well.) Parking can also be very difficult in Road Town, particularly during high season. There's parking along the waterfront and on the inland side on the eastern end of downtown, but if you're planning a day of shopping, go early to make sure you snag a space.