Several vendors sell tourist kitsch (sea shells, beads) on the Careenage at the Southern end of the Constitution Bridge (next to the Independence Arch).
There are numerous stores (including Cave Shepherd, the Macy's of Barbados) on Broad Street...especially for jewelry. Most of these specialize in duty free shopping for citizens of the UK, Canada, U.S. and others.
Swan Street, a pedestrian only mall, has stores selling cheap clothes.
The Number One Music Shop at the corner of Fairchild Street and Bay Street near the Careenage has a wonderful selection of Soca, Reggae, Calypso and other Caribbean music including local Bajan bands. Also a good place to pick up tickets for concerts.
Many duty free shops in Bridgetown cater to visitors, e.g., from cruise ships.
The small mall at the harbor also offers decent prices and selection, though goods produced in Barbados may be more expensive there than elsewhere on the island, e.g., rum. At least one department store in Bridgetown (Cave Shephard) offers a wide range of mercantile.
Barbados has a great variety of street vendors. Haggle aggressively. Don't stop until you're at about a third of the original price.