For many years I dove with Belize Diving Services. Have since switched to Paradise Down and now prefer them. email is firstname.lastname@example.org Their instructors are very good, prices are quite reasonable. You should be able to get certified in 3-4 days in that wonderful, warm, clear water.
There are quite a few good places to stay on the island. All depends on your price range and needs. Best to have a reservation, at least for the first night or two. Then you can look around and see whether you made the right decision.
For top notch service and spacious rooms I prefer Iguana Reef Inn. It is a bit pricey but you get what you pay for. www.iguanareefinn.com
Have a wonderful trip and welcome to the wonderful world of diving.