I'd second the Westin. We stayed there for 8 nights last August, though in just a hotel room, not a villa. Aside from the Westin and Caneel, most everything else is a rental, so I'd check VRBO, HomeAway, AirBNB, etc to see if something there meets your needs for price and space. The west end of the island around Cruz Bay is the most developed and therefore where everything is centered. There are some good places scattered around other areas, but for groceries and choices in restaurants, Cruz Bay is where to go.
Our favorite beach was Cinnamon Bay. They have bathroom and shower facilities and a small store and snack bar place that rents chairs and stuff for watersports. We spent two entire days there just sitting. It was fantastic. That being said, you can just hop to a different beach every day and have a blast. A Jeep is a must unless you are staying at the Westin, where taxis are always available. Taxis are basically old F-350 pickups or flat-bed trucks with benches on the back. It was $5 a head each way to and from Cruz Bay from the Westin.
Depending on your numbers, we took a day tour to BVI on
http://www.piratespenny.com/. They have two boats, each limited to 10 people so it isn't a ridiculous booze cruise, unless you and your 10 make it one. We stopped at The Baths, Cooper Island for lunch, and then The Soggy Dollar Bar. it was a great trip. You need passports of course, but you also need them to fly back to the mainland U.S. from St. Thomas (surprisingly).
We also took a snorkeling trip and sunset cruise through
http://blacksailsvi.com/, both of which were awesome. They aren't affiliated with a resort, they can do whatever they want. It's very laid back. I'd highly recommend them.
We also did a Reef Bay Hike through the Friends of the Virgin Islands National Park (
http://www.friendsvinp.org/news-a-events/event-calender/reef-bay-hike). You can do the hike without the tour, but if you do it on your own, it's an out and back that is all downhill on the way out and all uphill on the way back. Doing the organized tour, you only hike it one way (downhill) and then a boat picks you up and brings you back to park headquarters, and you get to see the south side of the island from the water on the way. Well worth it, and plenty of families on the tour. Not a strenuous hike when you are only going downhill.
Mr. Pipers Jeeps
https://www.rentajeepstjohn.com/ were great, just be sure to book as soon as you know your travel plans.
We had good food and/or drinks at the following, in no particular order. This is in no way an exhaustive or authoritative list.
De Coal Pot
http://decoalpot.com/st_john_location
Cafe Roma
http://www.stjohn-caferoma.com/
Vie's Snack Shack
https://www.yelp.com/biz/vies-snack-shack-coral-bay
Skinny Legs
http://www.skinnylegsvi.com/
Sam and Jack's Deli
http://www.samandjacksdeli.com/
Starfish Market (for groceries)
http://www.starfishmarket.com/
The Beach Bar
http://beachbarstjohn.com/