First, congratulate yourself on an excellent choice. You will find HHI to be a completely different experience than schlepping around parks. We love it at HHI, and we're heading back this summer for our 5th trip there. This is the vacation to seriously unwind, relax, kick back, and enjoy your family--no waiting in queues, no ADRs, no FPs, no crowded parking lots with a 15-minute hike to the park. Prepare yourself for the slower pace. Second, you're going to be amazed at the CMs at HHI. They are the best anywhere--friendly, helpful, and also very relaxed. Some consider them a second family. Now, onto the tips. These are just ours, and others will no doubt chime in with many others. Unlike at WDW, don't try to cram too much into your three days at HHI. The idea is to simply relax.
Food
Lots and lots of places to eat, for all manner of palates. If you want fresh seafood, and we always do, you have your choice. Our two favorite places for seafood are
Skull Creek Boathouse and Fish Camp on Broad Creek. Skull Creek does not take reservations, so either go early or be prepared to wait a spell. It has expansive outdoor seating, but it will be hot (though they have fans). For us, it's not a problem to go a little early and not have to wait more than 25-30 minutes, usually. They have the best low country boil we've found on the island. (There is also a Skull Creek Dockside virtually next door, and it's owned by the same company. We found the experience and food to not be quite as enjoyable as Boathouse.) For Fish Camp, they do take reservations, have outdoor seating with fans, and have a very good raw oyster bar along with very good service and a solid wine list. We've not had a bad experience at either Boathouse or Fish Camp. Right across the small bridge/entry to the Disney resort is Shelter Cove itself, and
Scott's Seafood Market restaurant sits there--maybe a 3 to 4-minute walk. It's not the greatest menu, but it's solid and very convenient. Right next to Scott's is
Hilton Head Social Bakery which is very, very good though pricey.
Ela's on the Water is also a good restaurant and is located another 4-5 minutes down the Shelter Cove shops area past Scott's.
If you want non-seafood options, there are a few places in the Shelter Cove shopping area which is less than a 10-minute walk from the resort along a shaded path. There you will find steaks, burgers, bbq, pizza, and even a stand-alone Starbucks. There are a couple of very nice Italian places within a 5-minute drive, too, in particular, OMBRA Cucina. Now, we often like to make our own dishes in the room, so we stop by Barnacle Bill's (5-minute drive) and grab fresh seafood, or barring going there, stop at the Kroger in the Shelter Cove shopping center which has a solid selection, too. We do our own boil! There is a QS at the main lodge near the pool, and while not an extensive menu, it's fairly good eats. They actually have fresh made Mickey waffles!
Activities
There are lots of daily activities you might enjoy. We're empty nesters now, so we don't do a lot at the resort, but there are tons of things for kids to do. Additionally, there are lawn games scattered throughout the resort: corn hole, putting green, giant checkers board, basketball, and shuffleboard to name a few. They have a campfire most nights; you can go crabbing for free with a CM right there at the resort; you can rent kayaks or bikes. Others can give you a better rundown on activities, I'm sure.
Beach/Pools
You have a pool and hot tub at the main resort (with a slide for the pool), and it's a great place to just relax and have a drink while the kids romp in the pool. There is also a pool at Disney's beach house, no slide. There is a snack bar there where you can get sandwiches and the like, and there's a place to get adult beverages, too. The beach house gives you private access to the public beach, though you can get to the beach without going into the beach house--there's a path right beside the resort. You can bike or walk to the resort, though it's about a mile and a half, so in the middle of the summer that can be very hot and steamy. You can also drive there but parking is limited, so either go early or expect not to find a spot without a little luck. You can also take the very convenient shuttle Disney provides that runs about every 20 minutes or so. For a family, I'd say that's the easiest way to get there unless you want to bike. If you've rented bikes, you can take the well-maintained bike paths that lead to the beach--they actually go under the busy main road (Wm Hilton Pkwy) so safety isn't an issue.
As for beach chairs, umbrellas, etc., you can bring your own or rent them from the lifeguards at the beach (private company not associated with Disney). If you don't bring your own stuff, then renting is likely the way to go. It runs something like $30 a day, I think, but they set them up for you and reserve them, so it's worth it we think. However, often we just go for an hour or so, thus take only towels and pay nothing.
There are plenty of other things about which to talk, but I'll stop for now. I know others will come on with even better tips and suggestions. My one overarching theme/tip, however, would be to not over plan the trip. Make it a time to enjoy sunsets, relax at the pools/beach, and spend time with each other. If you're there from mid-June through early July, there are weekly fireworks shows that you can witness from Shelter Cove or the marsh side of the resort itself. Find time to reconnect. I'm sure you'll love it there.