There are no non-stop or direct flights from the Washington, DC area to St. Kitts. You'll have to take a connecting flight. You best options are: 1) USAIR via Charlotte, American Airlines via Miami or Delta via Atlanta. Which one is best for you depends on day of week you are traveling as USAIR and delta usually only fly on weekends.

Other options are to fly from DC to St. Martin or Antigua (if you can do that non-stop) then take either a Winair or Liat island hopper flight to St. Kitts but keep in mind that the local commuter airlines don't always keep "on schedule" so it's best to connect through one of the US airports I mentioned and fly from either of them non-stop into St. Kitts.

As for what to do/see on St. Kitts consider visiting the following:

The Fortress at Brimstone Hill
The Batik Factory & Gardens at Romney Manor
The Sugar Train
The Petroglyphs
The Black Rocks Lava Flow
Go Scuba Diving
Do a day sail snorkeling trip
Visit Nevis for a day