We own a home on the island, and have been going there as owners and renters for the last 20 odd years, so yes, I guess I know a little bit about what to do.
Food: There are Publix grocery stores, Harris Teeter which is a bit more "upscale" and a new entry, FreshMarket. I recently had an experience with the management of FreshMarket which was so unsatisfactory that I called corporate, so I cannot endorse them. However, they do have an interesting inventory of items not generally found elsewhere even off the island.
What to do: Obviously, there is the beach. And bike paths and water activities. Tennis at many locations. Miniature golf. No one in my family golfs, and there's plenty to keep us busy. There are numerous places that rent kayaks, and some organized kayak outings that will allow you to explore places where you might actually see interesting things. There are dolphin cruises in lots of variations--my favorite is Commander Zodiac which takes you out in inflatable rubber crafts which seat about 8 people. Pick up an Island Events magazine as you check in--lots of information about what's happening and who/what offers various activities. Hilton Head isn't like many of the East Coast beach cities which function around a boardwalk, but there are several resort type commercial areas...Harbortown, Shelter Cove, South Beach (which is where our house is) all have both dining and shopping as well as some entertainment activities.
Much of the island is made up of private, gated communities/resorts. These have various entry requirements. For example, Sea Pines can be accessed by those not staying within it by paying a daily gate fee. Both Harbortown and South Beach are in this community. Shelter Cove is open access, but the resort which it is part of is not. Hilton Head Plantation requires that you be staying there or have a reservation at a restaurant within its gates. Most others require the same. An obvious exception to this is when an owner calls in a pass for you to pick up. There are lots of commercial areas outside plantations, however, so you can certainly fill your time without entry to these areas.
Restaurants: There are over 300 restaurants on the island. Most are fairly casual, and most cater to both locals and tourists. The big difference is that owners have figured out that you have to call ahead for reservations during tourist season. I would be better able to help in this if I knew what types of food you would be most interested in, and the price range that you expect.
Sightseeing: Our fulltime home is just outside of Washington D.C. In comparison, I'd have to say that HHI is not a sightseeing stop in the traditional sense. It's much more a kick back and relax venue. For sightseeing, however, both Savannah and Beaufort are short day trips. We finally "did" Savannah two weeks ago and it was really nice. Lots of options for companies and also walking tour maps for sale.
