That's rather a big question and not one that is easy to answer.
With only nine days, you can't see all of Belize or even a good part of it. Though Belize is not a large country, getting around takes longer than you think. I would say to pick two areas, three at most, and spend your time there. Probably most people would pick one of the cayes and Cayo district for a first trip.
For budget level travel, for the island part of the trip I'd say either Caye Caulker or Tobacco Caye. The rest of the time you could spend profitably in and around San Ignacio. There are budget places there (among the best: Aguada, Trek Stop, Martha's) and lots to do -- caving, hiking, canoeing, cave tubing, visiting Maya sites, etc.
Belize books and publications by Lan Sluder:
Belize First Magazine (http://www.belizefirst.com
Fodor's Belize & Guatemala Guide
Belize First Guide to Mainland Belize
Adapter Kit: Belize (the first and only comprehensive guide to living, retiring, working and investing in Belize -- a best-seller on Amazon.com)
UpClose Central America
San Pedro Cool (coming soon)