We went to both of these places on our first cruise with Royal Caribbean. We found Martinique to be filthy and smelled like raw sewage when leaving the ship. We found out why when the cab drivers just walked over to the waters edge and urinated. One couple even saw a little girl squat and urinate in the street. After seeing this we decided against going on any excursions here. I hope they have cleaned up their act. I know RCI no longer stops here. As for Barbados we LOVED it. We took the excursion offered by RCI that took you to the beach and a tour of the rum factory across the street. The beach was a very secured area, they had policemen patroling the beach area as well as the front by the street. We were not hassled by people trying to sell us things. They had bathrooms and showers here and the excursion came with a beach chair an umbrella and a drink. I don't remember the name of the beach, but it wasn't too awfully far from the pier. I don't know what Carnival offers here, but I can highly recommend this beach.
Can't speak to Martinique - but Barbados was the best port of call on our March cruise. We hired a private guide for the entire day and he took us to all the scenic places on the Atlantic Coast - so much more quiet in comparison to the Caribbean Coast of the island and not nearly as built-up with modern amenities so you get a better sense of the island itself. We started at a plantation house, and then went on to Bottom Bay (beautiful, white sand beach, but the surf was way too rough that day for swimming), Ragged Point, St. John's Church, Bathsheba, and a whole range of other places. We'd get out, wander at our own pace, and return to the van when we were ready.
We had two goals for this excursion - see and photograph beautiful places and then visit the wildlife preserve at feeding time to see the monkeys - an absolute must do in my opinion. Both were accomplished to our complete satisfaction. Our only regret was that we had just one day to spend on the island.
The tour company we used was Boyce's Tours - they also do tours for cruiselines, so they came well referenced - website is www.boycestours.com.
DO NOT miss Harriman's Cave (or is it Harrison's?) while in Barbados. It's one of the largest caverns in the northern hemisphere, so big that you ride in a TRAIN wearing a hardhat and it STILL takes 45 minutes to see it all. I have been through caves all over the US, but we were blown away by that one.
Also, be sure to sample their national dish - flying fish! - while there, at a zillion restaurants all over the island. They even have flying fish on their coins.