Lake Nicaragua spreads out in front of you like a fresh-water sea. On especially clear days, you can see all the way across the lake. You can swim in the calm water, relax in a hammock on the beach and have a drink at one of the surfside restaurants. From Santo Domingo, you can find a guide to take you by horseback to see the petroglyphs for which the island is famous. Unfortunately guides are not always particularly well-informed about their significance.
Horseback tours are available as well.
You can also arrange for a guide to hike to the top of the Madera Volcano. The town of Altagracia on the northeast side of the island offers a feel for traditional Nicaraguan culture. You might also consider renting a bicycle for a few hours to ride around the area. The town also serves as a good starting point for the daytrips listed above. As the island is primarily agricultural, you can expect to see rice, beans, corn, bananas, plantains, and other crops.