There are a couple of DHHIR lovers and/or owners threads on this very forum (
DVC RESORTS) which might have some good suggestions for you. Further, there is fb page that is strictly related to the resort, and it has lots of posters with in-depth recommendations.
One suggestion is to do a dolphin cruise. A few outfits do them, but the one closest to the resort has an excellent reputation: Outside Hilton Head. There are plenty of historical sights to see in HHI, many of them related to the Gullah culture here. Savannah has more and is an hour's drive; Charleston has plenty to see but is two hours away.
As for restaurant recommendations, everyone will have his or her favorites. For the most part, you can't go wrong with most eateries here--as a whole, they are way better than Disney dining options and less expensive to boot.
Keep in mind that Easter on the island is extremely busy and many popular restaurants do not take reservations. Further, my list does not include any in Sea Pines as entrance to that part of the island (about four miles past the resort) is $9 per day per car. There are so many great places to eat outside of Sea Pines that we see no need to pay extra. Still, here are my favorites in no particular order.
Fishcamp on Broad Creek: Excellent food, good ambiance, outdoor seating available, and they DO take reservations (they use Resy). Approx. 10-12 minute drive from the resort.
Ela's on the Water: Also topnotch food and ambiance with outdoor seating, and they, too, take reservations (you can use OpenTable). Approx. 10-12 minute
walk from the resort (it sits in Shelter Cove).
Skull Creek Boathouse: More of a local feeling place but it is almost always crowded. They do not take reservations. Extensive menu with one of the top two or three low country boils in the area. Fun place to eat with a huge outdoor seating area (inside is smaller). Still, did I mention it gets crowded. Wait times on weekend evenings can be well over an hour even showing as early as 5-5:15. Drive time is about 15 minutes.
Hudson's on the Water: Another local gem but quality is sometimes hit or miss (mostly hit). Again, lots of outdoor seating on the water but similar to Skull Creek Boathouse, it is almost always busy and wait times can be astronomical. Located in same area as Boathouse, so about 15-minute drive.
The Black Marlin: Smaller than either Boathouse or Hudson's, the overall quality isn't quite up to par with either but it's usually darn good, and it's easier to get into, as well. OpenTable is available, and it's about a 12-15 minute drive from the resort. Outdoor seating available.
Scott's Fish Market: Sits right across the bridge from the resort, and it's decent. The best component to Scott's is you can get there in about 5-7 minutes walking so it's very convenient; however, I don't believe they take reservations and we find the food just middling...except for the crab cakes where are excellent. Plenty of outdoor seating.
Hilton Head Social Bakery: Sits right besides Scott's and is the best place on the island to go for French bakery items such as baguettes, quiches, macarons, tarts, breakfast and lunch items. They aren't cheap, but they are exceptional. There is almost always a line outside the door, so prepare to be patient unless you hit them when they first open.
OMBRA Cucina: Best authentic Italian on the island and a little more upscale though not pretentious in any way. Great wine list, menu, and service. It's a mere 5-minute drive from the resort. Being upscale, you'll pay a bit more there than, say, at Frankie Bones for a similar dish, but the quality is notably higher at OMBRA. Reservations recommended.
Michael Anthony's: Also topnotch Italian, but not quite up to OMBRA...close, though. Friendly staff and great takeout menu. Located about two minutes past OMBRA. Reservations recommended.
Charlie's L'Etoile Verte: Upscale--again not pretentious--fusion of French and Low Country vibes. Reserve early as it's quite popular as one of the best spots in HHI. Very good but not as amazing an experience as others tend to gush about it. It is also about two minutes past OMBRA.
Alljoy Donuts: This place is located near downtown Bluffton, so it's a good 20-25 minute drive, but they have the best donuts around (sorry Purple Cow and Duck Donuts--you can't hold Alljoy's yeast). They are only open Fri, Sat, and Sun from 0700 to 1300, but they are worth the drive. A simple glazed donut says all you need to know about a donut place, and theirs are marvelous. Other favorites include the Reese's PB Cup donut, maple bacon donut, and currently their cranberry-orange donut.
That's it from me. Don't tend to eat BBQ or burgers or pizza except at home, and based on my experience, most of those type places are of similar quality--pretty good to good. Good hunting!