Depending on where your apt. in Speightstown is located, you might be able to walk into the town. There are excellent restaurants along their boardwalk such as Fisherman’s Pub, Orange Street Grocer. There is also Island Plates and a new one just opening called Hugo’s. Also about 10 min. away by bus or taxi is The Fish Pot restaurant in an old fort, right on the water - excellent food and you can also use their beach and sunbeds. There is an excellent interactive museum in Speightstown, called Arlington House, depicting the history of Speightstown and Barbados.
Contrary to what was mentioned above, The Fish Fry in Oistins takes place every day of the week. If you like alot of people and activity, go on the Friday, otherwise any other day of the week is fine. There is usually some local entertainment happening there all the time. Oistins is a fair distance from Speightstown, you could take the big blue government bus to Oistins and then a taxi home. There is also an excellent fish fry that takes place in Moontown, which is closer to Speightstown.
From Speightstown, a short local bus ride away, is Holetown, which has an upscale shopping mall, ‘Limegrove Mall’, with some good restauraunts such as Lemongrass or Relish, and the rooftop Fusion Restaurant & Bar. Also right on the beach in Holetown is Surfside Restaurant for great local food, as well as Zaccios.
You do not mention your interests, but there are many things to see in Barbados. Orchid world, Hunte’s Gardens built by Anthony Hunte in a huge sinkhole where you can wander around, and even bring a picnic lunch or just a snack and something to drink and spend time in this beautiful place. If you are lucky Anthony may be around and he is always willing to speak with visitors to his huge garden. There is the newly renovated Harrison’s Cave, with spectacular views and a nice restaurant for a quick lunch.
Visit the East Coast and Bathsheba, and walk along the long beach and see the huge crashing waves. There is also a restaurant there at Barclay’s Park, lovely to have lunch on their deck.