I've spent 4 winters in the Eastern Caribbean and have seen most of it. Faster and Mark in their posts above offer good suggestions. Antigua is a great island with the best parts IMO being English Harbor (home of Nelson's Shipyard) and Nonsuch Bay, a fairly remote spot on the windward side sheltered behind a reef. The French islands are nice and very cosmopolitan by Carib standards. Our favorite spots in the French islands include Iles de Saintes and some of the bays south of Fort de France on Marintique. If you're looking for some city life Fort de France (Martinique) and Point-a-Pitre (Guadeloupe) are good choices, but are lacking somewhat in "island" atmosphere. Sint Maarten (Dutch)/St. Maritin (FR) is a good stop and as mentioned the marine trades available there are very good. The Dutch side offers easier anchoring (Simpson Bay Lagoon) and is a bit over-developed IMO having much less charm than the French side. There are places in Sint Maarten where you think you're in the US.
Re routing for young travelers from Wyoming -- they will have to change planes somewhere. They can probably get direct flights to many major islands from Chicago and many more islands from Miami / Ft. Lauderdale. Flying through San Juan presents even more island destination options with the major US airlines and is an easy connection to places like the US and BVI. Islands where it might be difficult for you to get to the airport to pick them up would be anywhere in St. Vincent and the Grenadines, Dominica, the airport on the south side of St Lucia, anywhere in Martinique except Marin and Fort de France, anywhere in Guadeloupe except Point-a-Pitre and anchorages on the east and south coasts of Puerto Rico. Among the easiest places in the Caribbean for meeting arriving crew are St. Thomas, BVI (Beef Island), Sint Maarten, Antigua, Point-a-Pitre, Fort de France and Castries in St. Lucia.
Although probably not your first stop in the Caribbean, I encourage you not to overlook the US Virgins, which are a great place to have kids visit -- and especially St. John. If I were picking one place for grandkids to visit and I wanted to be sure they had a nice tropical experience, it would be St. John. You'll will find very few places in the Caribbean prettier and more restful than the bays on the north side of St. John and few places as unspoiled as those on the south side of the island. All the major bays on St. John are in the national park, all have high quality moorings at reasonable prices ($15/night), most have great sandy beaches, there are great walks to take ashore and, best of all, much of the island is still in a natural state. On the other hand, St. John is American, tropo-American yes, but you know you're in the US. If it's the true Caribbean cultural experience you want them to experience you can hop over to the BVI. Downside of the BVI is all the charter boats. Connections into and out of the US/BVI easy via the San Juan and there are more and probably better opportunities for varied day sailing in the US and BVI than you'll find anywhere else (except perhaps the Grenadines).
Stopping in the VI also sets you up nicely for the trip to Panama. The passage from the VI or eastern Puerto Rico to Colon is not difficult -- 6-7 days in the trades. Starting from there (vs the Windwards) shortens the passage somewhat and gives you an opportunity to avoid the windy zone usually found 25-150 miles off the Columbian coast. This is an area where the winds are sometimes near gale force and the seas are among the largest in the entire basin. (See the Caribbean pilot chart). If you were to take a course that runs from the USVI to about 14N 77 30W and from there direct Colon you can avoid the area mentioned above.
Good luck and have fun with the grands.