Getting to Anegada requires some effort, which is most of the point. There are two options from Tortola.
Ferry: The Speedy's ferry runs from Road Town, Tortola to Anegada several times a week — check the current schedule as it changes seasonally. The crossing takes about 90 minutes.
Fly: Cape Air and other small charter operators fly from Terrance B. Lettsome International Airport (EIS) on Beef Island, Tortola to Auguste George Airport (NGD) on Anegada. The flight takes about 15 minutes and the views over the reef are remarkable. Charter flights from the US Virgin Islands are also possible.
Day trips from Tortola are possible but barely worth it — Anegada rewards an overnight stay. Most visitors stay for two or three nights.