You will see some of the National Park on many tours to St. John. There is no charge to enter the park since about half of the island is part of the National Park, including many beaches. Trunk Bay does collect a daily fee, however.
St. John is my favorite island in the Caribbean. Unfortunately it is difficult to do this island justice in the short time that the ship is docked at St. Thomas. You'll have an approximately 45 minute ferry ride each way.
When on board, visit the Shore Excursion desk to have your questions answered. The staff is phenomenal at suggesting the best excursions given your interests and the ages in your traveling party. If you venture on your own, you'll need to budget your time wisely in order to get back to the ship in time.
You may want to check out the
St. John Eco Hike. The St. John Island tour will take you to Annaberg Ruins which takes you through portions of the park!