For general grocery items, the cheapest supermarkets are People's and Bottom Dollar, but always check the packaging and "best before" dates. Be careful with refrigerated/frozen items. Save U has the best selection, but it's probably the most expensive supermarket in Belize City.
Buy fruits & vegetables at the main market beside the bus terminal.
For fish, buy direct from the fishermen at the fish market.
For household stuff, lots of small stores on the streets in the downtown area of the City -- Albert St., Orange St., Queen St.
For clothes, spend a day in Chetumal.