Many of the Caribbean islands are sovereign nations, others are still parts of France .. in any event for the most part every time you change islands you'll be needing to properly 'clear out' and 'clear in' at the next port.
Some islands are fussier than others, but somehow you need to get on to the required protocols. Ignoring or violating their rules can result in serious problems. I think Noonsite is one area that has a lot of info.
The Doyle guides are used by most in the Eastern Caribbean: Doyleguides site lobby
We have sailed a bit with friends who winter there (essentially what you plan to do) - this is their 10th season. They are very careful not to alienate the locals... patronize the 'boat boys' to some degree (but don't be taken advantage of), they don't 'go into town' in swimsuits - many locals dress very nicely day to day.
There's a definite different flavour between, say, Guadeloupe (a department of France) and Carriacou or Bequia.. Saba is a world of its own. The cruise ship stops are very different places when the tourists come to town...
Island hopping is a great way to go.. daysails of 40-50 miles in the eastern chain with reliable winds, mostly reaching at good speeds. For the most part reasonably good anchorages abound.
For us, (we've covered most islands between Grenada and St Maarten) favourites include Guadeloupe's Iles de Saintes, Bequia, Tobago Cays and Carriacou. Saba was special in many different ways but it's not an easy stop - no real anchorage and no real shelter from swells. Still in the right conditions it's a memorable visit.