Vieques is a small island off the coast of Puerto Rico, on the boundary of the Caribbean and the Atlantic. To get to Puerto Rico, fly from almost anywhere in the world to San Juan International Airport (SJU).
Puerto Rico is lies between the Dominican Republic and the British/US Virgin Islands. Vieques is sometimes referred to as being part of the Spanish Virgin Islands.
Politically, Puerto Rico is a Territory of the United States. US citizens can travel here with passport or drivers license and birth certificate. Resident Aliens of the US should bring their green cards and/or passport with visas. Non-US citizens should find that entry requirements are basically the same as if you wanted to enter the US.
From San Juan airport you can either fly or take a taxi/ferry to Vieques. SJU to VQS is a 20 minute flight. The taxi drive to the port at Cieba is about 40 miles (an hour or so) followed by a ferry ride of about an hour. You can also take a taxi to the airport in Ceiba and take a 10 minute flight from there.