Just off the coast to the east are numerous idyllic small islands called cayes (pronounce keys). South Water Caye, Tobacco and Laughing Bird Caye, each a beautiful little paradise are a wonderful place to spend a few days chilling. You can snorkel right off the beach on many of them. Depending on your budget you can find a range of accommodation from high end to camping. If you are looking for more hustle and bustle, there is always Caye Caulker and San Pedro to the North of Belize City.
To the west of Hopkins lie the Maya Mountains covered in thick rainforest. There are a number of day trips that can be made into this environment, depending on your style. Just South of Hopkins is the Sittee river and paddling a canoe along it can be a sublime experience. Lush jungle lines the river with parrots, toucans and many other birds easily spotted along the way. The Cockscomb Basin Wildlife Sanctuary and Jaguar Preserve is a fascinating place to visit as well. If you are feeling particularly fit and adventurous, you may want to hike up the Mayflower waterfalls just west of town in the mountains. All of these activities can be done as part of a tour or arranged on your own. Just ask Dorothy at Lebeha Drumming Center for more details and she can either book or help you to arrange something.
Belize has its very own Mayan ruins in the Cayo district near San Ignacio. They are quite beautiful and an excellent alternative if you can't fit in a trip to Tikal in Guatemala. It's best if can spend a few days in San Ignacio where you can easily arrange a day tour.