What is the best time to visit

It really depends what you want to do--if you are more adventurous, inland San Ignacio or up near Hidden Valley Inn & Reserve can be a great place for hiking, wildlife, etc.  If you like archaeological sites, Caracol is definitely fantastic. 

For snorkeling and/or diving, Ambergris Caye, Belize is definitely the most popular. While the beaches are not particularly picturesque, the undersea wildlife and coral are fantastic. If you aren't a diver, don't worry, there is a ton to see while snorkeling too. There are plenty of different style resorts and hotels to choose from, whether boutique luxury, budget, or classic hotel style.

I think you can visit Belize anytime. Inland it can rain frequently, so grab a jacket. It can be hot May to September both inland and on the coast, and I think April is considered the real "dry" season.  In December and January, it can actually be cool inland (not chilly, but not shorts/tank top weather either).