You may be able good deals, but you need to be an educated buyer.
Do some shopping at home first to see what prices and quality you want. You need to have some basis of comparison to see if you are getting a good deal. Things to research at home: stone cost vs size, color and quality; metal costs vs alloy mix and weight; construction technique and craftsmanship.
You mentioned you wanted diamonds. Get a diamond tester. You can get good handheld battery powered ones for about $40.
In our experience looking for tennis bracelets, we found that the value of the finished products at DI were lower than what we could get at home. The quality of construction was the biggest problem. The settings and links were very poor quality and weak compared to bracelets from reputable jewelry stores at home. The stones were ok, but in the end, the overall value was not there for something we want to last for decades. We talked to our local jeweler about this and they told us we may be able to get good prices down there, but there's a reason why the prices are lower. Make sure you understand what the reasons are before you buy.
I would recommend buying something that is not made there. Buy things like watches or brand-name jewelry (like Chopard). That will take the quality issue away and make it easier to compare costs--no taxes down there!
As for where to go, I agree with a PP about Grand Cayman. Also, if you want to go to DI, don't go to the store at the port, walk down the road to their other DI store by the Atlantis submarines place. The products at the non-port store seems to be nicer and handled by fewer people. Also, all the stuff we were asking about seemed to be at the non-port store and they kept sending someone to fetch it. We ended up just going to non-port store to look for ourselves.