Take a full day island tour with Antigua Premier Tours. They are willing to make recommendations and take suggestions on things to see and do and can pre-arrange lunch for you, too.  You will be met at the dock and have a great day touring in the air conditioned van. They also provide local snacks and drinks for you all day.

You can also check out TripAdvisor's Antigua pages or the WCities website has a list of more specific places and attractions in Antigua that you can see on your own on foot, or take a tour.