where to go grocery shopping that is reasonable - Just outside of the Westin entrance is a "new" grocery store. It was constructed but not stocked when we visited last. I have read good reviews from here and other forums about the new grocery store.
We have always used Starfish market which you will find is on the way to the Westin from the ferry dock. You should be able to find anything you need at the Starfish. The prices are 10% to 30% higher than mainland prices with most items falling in the 15% -20% range.
best places to eat - You have a number of choices for eating out. YOu can get anything from BBQ & Barfood to fine dining in and around Cruz Bay.
some of the best activities to do there - Will you rent a jeep while on island? If so your opportunities for hiking, snorkeling, beaches are virtually limitless.

Since this is your first visit I will suggest that you make a stop by the website www.seestjohn.com. Also consider purchasing a guide book to find out a bit of the islands history as well as offer great ideas about what there is to do and where and how to do it. You can find guidebooks on VINOW.com or the seestjohn.com websites I recommend the book Off the Beaten Track by Gerald Singer. (same Gerald Singer who manages the www.seestjohn.com website) Your biggest activities available are hiking the National Park Service trails (since 2/3 of the island is National Park), snorkeling, some windsurfing, some kayaking, dinghy rental and most of all sitting on a pristine beach with a nurturing caribbean sun on your shoulders.

The St John Market that Toes mentions is open and has become one of our favorites for shopping. It is clean and well stocked and the staff was super helpful.

My favorite reasonable restaurant is Banana Deck. There are so many great choices all over the island. If you have a jeep rented, head over to Coral Bay and check out Skinny Legs or Shipwreck Landing.

I recommend Feet, Fins, and 4WD. It is a great guide to the island and can be purchased at www.caribmart.com
There are so many hiking trails, if that is your thing.
We are beach bums, so we love to just chill on the beach all day.
Cruz Bay is a great town to roam around, have a drink, do a little shopping, and people watch.

It sounds like you are staying in the Westin Villas, correct? Good decision if you are.

The Westin has a lot of on site activities to keep you busy if you choose to stay on site. The beach there is nice, but the water is not like some of the crystal clear snorkeling beaches on the North Shore. I recommend taking a taxi to a couple of the North Shore beaches and try snorkeling. If you are just a bit more adventurous, rent a jeep and explore the Island and beaches yourself. There are daysails available right at the Westin which will take you snorkeling and to Jost Von Dyke in the BVI's, just bring a passport.

The Westin has a good steakhouse, and some other restaurants on site including a poolside bar and burger type food, and a deli/grocery store. It is very expensive to buy anything at the deli so the grocery store just down the road is a better option. I usually bring some high quality steaks down with me and grill them at the Villas nice gas grill/pool area. I go into town and eat apps and have a drink or two at some of the great restaurants. I like Lime Inn, Waterfront Bistro, Beach Bar and Woodys, JJ's, Morgans Mango, and Banana Deck.

The ranger station in Cruz Bay offers reasonably priced trips down Reef Bay trail to the petroglyphs with a boat ride back to Cruz Bay. Lots of good food places to choose from in Cruz Bay as well, but I like the local chicken place across from Cap's (Uncle Joe's? might be the name of it). Skinny Legs over on the other side is also a good choice. Brunch at Caneel Bay can be a good adventure. Snorkeling at Hawk's Nest and at Waterlemon are favorites of mine. Depends on whether you'll be renting a car or using taxi's. Westin is supposed to be beautiful, so you could probably spend most of your vacation there and at Cruz Bay and be pretty happy. Enjoy!

for dining, i recommend the lime inn (all you can eat shrimp on wed, i think), woody's (bar food), skinny legs (great burgers and key lime pie), shelia's pot (great local food and cheap, by ferry dock) and uncle joe's bbq (awesome ribs and chicken, near ferry dock).

take a day trip over to jost van dyke and visit foxy's and the soggy dollar!

Thank you all for the helpful posts.