Belize is great for fishing nearly all year round. You can go after tarpon, snook and jacks off the beach or river mouths. Bonefish and permit are common on the grass flats and the reefs are home to snapper and barracuda amongst others. Although I was there for fly-fishing, I had a great day trolling for barracuda. Tarpon were a lot of people's primary target and there were some whoppers around. See the photo below of one I saw whilst diving. In 2007 they were relatively easy to catch.
Belize is home to numerous islands and Caye Caulker and Ambergris are two popular destinations. I stayed on Caye Caulker which has a charmingly ramshackle feel to it. There are plenty of restaurants and bars although in 2007 there was little in the way of international hotel chains. Probably for the best. Along the main street in Caulker were a number of fishing shops where you could hire tackle and guides to take you out were easy to find. Peak times for fishing the local flats are May to November but regardless of when you go one of the guides will help you to catch fish.
If you are there with your family and can't bring your stuff or have the opportunity to take yourself away for a whole day, all is not lost. Just hire the gear from one of the shops, stick on an old pair of trainers for wading and you'll have everything you need to spend a few hours fishing around the island. You won't have to walk far to find somewhere half decent to fish and although you won't have the success you would if you took one of the guides, you'll still have fun. I caught a number of small jack (with a Snappy Shrimp) doing just this.
Getting to Caye Caulker is easy. A flight to Belize City, taxi to the port and finally a water taxi will have you there before you know it. If you go it would be worth trying some of the other activities around the island. You can take classes in windsurfing or if you are a diver you shouldn't miss the opportunity to dive the blue hole. Again there are numerous operators that will take you out. If you don't dive I would suggest taking one of the snorkelling tours to see the rays and tropical fish.