Austin Bergstrom International Airport or airport code is AUS
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While AUS is the only airport in Austin. San Antonio international (SAT) is also a short drive away from Austin. Also if your up for a drive the Houston has 2 airports IAH and HOU. That is about a 2.5 hr drive.
Most popular airlines that fly into AUS are Southwest Airlines, JetBlue Airlines, American Airlines, Delta, and United. There are some additional airlines that come into Austin also. So you can look to https://www.google.com/flights to begin. But you can go to each airline website to check flight. Also be open to traveling with different airlines to get to Texas. For Instance JetBlue has a direct flight from Barbados that fly into Fort Lauderdale FL (FLL). Then there is a direct flight from FLL to AUS with Southwest and JetBlue. Note for NY travelers: There is a direct flight from JFK to AUS to leaves at approximately 9:00pm. Note for California travelers: Both Southwest and JetBlue has direct flights from LAX and LGB Note for St. Lucia travelers: American Airlines may be the best bet from UVF to MIA to AUS. Note for STX travelers: You have many option but a more direct flight would be with American Airlines STX to MIA to AUS, or Flying with Seaborne to SJU and connect.
While the weather is Texas in unpredictable one thing you can count on is lots of HEAT. So travel with light. Most activities will involve casual clothing.