Congratulations on your new home! It really depends on your budget and taste. We had some furniture made by the excellent carpenters here, brought a few pieces from the UK and bought some at Mirab furniture store.
Mirabs is in Belize City. You should also look at Hummingbird. They have rattan, bamboo, wicker and hardwood. They have a showroom on the island, but have a lot more at their larger factory and showroom in Belmopan.
Graniels did some lovely work for us, but we didn't use them exclusively although I must say that the quality of their work was fantastic and we enjoy the work they did for us very much. Have fun shopping around. Mirab sells more conventional furniture and is indeed in Belize City.
If you are thinking of having anything with slip covers or chair pads etc. I would highly recommend Sunbrella. It will hold up to the sun and elements very well. Kate at "Sew What" has a very nice selection and does excellent work. Emma Carter has made many draperies and pillows for me and is very good.