Deep into the Sierra Maestra mountains (near Bayamo) you can walk the trail to the historical site of La Comandancia de la Plata. From Bayamo you'll need a car or a taxi to come to Santo Domingo.
Start early in the dry season (from november till april) and beginning of the rainy season until end of June. Shoulder season May and June, showers may occur. July and August are the Cuban school holiday.