Have lunch at the second-story Hemingway's Restaurant. If you enjoy people-watching, get a table on the verandah overlooking the street. Don't look for Papa Hemingway. The best part of the island is outside St. John's. One day will provide a sampling, but a two-day excursion is better.
Distances are short, but stops are frequent.
Drive your own rental car, or negotiate a tour by taxi. It's not as expensive as you might think. Here are some suggestions, moving roughly clockwise around the island. Drive east from St. John's to Parham which was settled in the seventeenth century. The octagonal St. Peter's Anglican Church, built in 1840, is important in the ecclesiastical and architectural history of Antigua.