In Barbados definitely go to Oistins, which I believe is actually in Christ Church, and enjoy the fish market. Many stands serving grilled fish and other local specialties. After trying a few of them we settled on the one called Mo's Kitchen. They actually grill their fish well as opposed to burning it over open fire, and their portions are huge. The BDS$25 special they ran while we were there literally fed a family of four.
Closer to Bridgetown there's a place called Eve's Garden that's a very good option for local food too. It's more of an upscale sit-in type place, but still affordable.
There's several roti shops and other good places to eat along Hwy 7. I actually created a little Google map showing most of the places we went to in Barbados.