my instinct on reading your request is grenada. do some reading on it plus guadeloupe and st. lucia. a good start is to just type island name into search box above. you can get a feel for each island by reading ppls reports and answers to questions.
st. lucia is huge and ppl often get from one end to the other by helicopter. it has zipline, only drive thru volcano on earth. not known for snorkeling or it's beaches however.
grenada & guadeloupe would have the carib vibe you seek with nice beaches, some snorkeling & markets. french is useful.
the more specific you can be about your budget the easier it will be for us to help you find a resort.
the best snorkeling is on the flat, arid islands like grand cayman or turks & caicos.

Unlike Hawaii, where all of the main islands except Moloka'i and Lanai have TONS of activities going on, most islands in the Caribbean that offer all of what you're asking for would also happen to fall under the category of "touristy" in my book: horseback riding, helicopters, zipline, etc. Also, the lusher, mountainous islands tend to have the fewest white sand beaches because they're volcanic. It won't be impossible to find much of what you're looking for in one location, but you might not be able to find it all.

I think virginia's recommendation of Grenada is spot-on. It's a large enough island with both touristy and not so touristy areas that's lush & mountainous with a few white sand beaches. It's usually expensive and a long travel day to get there (not as long as Hawai'i, probably!), but once there, you have a pretty wide range of prices for food, accommodation, etc. It's one of my two favorite islands and you can find almost everything you need to know at www.grenadatravelforum.com and www.grenadaexplorer.com. It has a great West Indian vibe and it might feel less touristy to you because not many Americans vacation there.

st. Lucia also might work for you, but it's a bit of a catch-22. The parts of the island that are developed are VERY touristy and the more remote areas might not offer enough of what you're looking for. But like Grenada, it's lush and mountainous. No natural white sand beaches, but there are a few with light sand and there's one resort beach that has imported white sand from Guyana for its guests.

I would also recommend Grenada as one of the more non-commercialised islands. It has a great range of accommodation including 5-star resorts, hotels, villas, cottages, self-contained apartments and guest-houses. It is known as the Spice Island and it also has a great rainforest with a lake at the centre; lots of waterfalls; great beaches (some are secluded and others are more public); good snorkelling and a visit to the Underwater Sculpture Park for a snorkel should not be missed. Take a look at the official website of Grenada and see what else there is to see and do...www.grenadagrenadines.com.

I too would highly recommend Grenada. My wife and I have vacationed there twice and would not hesitate to return. In my trip report I have provided a link to a video regarding the Underwater Sculpture Park.....it really was quite amazing.