You're going to love it here. This resort is the place to simply do one thing: r-e-l-a-x. This is not your typical rush-rush, ADR, FP, park-hopping madness you encounter at WDW.
As for suggestions, here are a few, and I'm sure others will chime in as the days go by.
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Rent bikes. This is the easiest way to get around the island. There are over 100 miles of bike trails, and they'll even take you to the beach (you go under the main thoroughfare). I suggest renting at the resort as the price is virtually the same as offsite rentals, and if something goes wrong, the bike shop is right there at the main pool.
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Hang out on your balcony. Whether it's early morning coffee/tea, iced tea at noon, or a favorite libation in the evening, sitting out on your balcony and soaking up the ambiance is a marvelous way to unwind.
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Cook in the room. Yes, the 1BRs, 2BRs, and GVs have pretty much everything you need to prepare a meal, and as mentioned, if not, call the front desk to see if they can accommodate a request. Most typical items they will have available to send to the room. If not, there is a huge Kroger's literally moments away in the Shelter Cove Center. You can buy all your groceries there, too. As for fresh seafood, I highly recommend Bennie Hudson's which is about an 10-12 minute drive from the resort or Barnacle Bill's which is only 4-5 minutes away in the same direction. In fact, you pass by the turn to Bennie Hudson's on your way, and you drive right past Barnacle Bill's minutes later on the right.
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Breweries. There are three or four in Bluffton (the community you drive through before you reach the island) and one on the island. I've heard good things about them all.
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Mini golf. If you enjoy this sort of activity, there are a few on the island, and you can bike to them all.
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Kayaking. Outside Hilton Head operates out of Shelter Cove, so you can walk there in just mere minutes from your room. They do excellent guided kayak tours. And if you want to do a dolphin tour, they provide those as well. For the record, you may very well see dolphins if you go kayaking. In fact, OHH does a lot of charters from fishing, to harbor tours, shrimping, paddle-boarding, and more. They have an excellent reputation.
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Eating. The low country vibe is all about good food, and there are so many great places to eat in HHI. Unlike at WDW, you needn't worry about making reservations way far in advance, though some of the more popular places may be difficult to reserve at your preferred time. One of the best spots, Skull Creek Boathouse, doesn't even take reservations, but they do have a huge outdoor seating area. Whatever you like to eat, you'll find it here: BBQ, French, Italian, Greek, American favorites, and of course, seafood--fresh seafood.
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Miscellaneous.
1) Yes, the resort provides pool towels both at the main pool and at the Beach House pool--they are the same towel. While not as big as a beach towel, you can absolutely take them to the beach, just be sure to bring them back and put them in the towel return at the Beach House, though you can also take them back with you and either put them in the towel return at the main pool area or even leave them in your room.
2) The only available food options at the resort are Tide Me Over which is open 8-4 and is a QS (they make fresh, made-to-order Mickey waffles among many other items). At the Beach House, the Surfmen's Sand Bar is open daily with snacks and drinks available; currently, from Friday-Sunday, you can also order typical QS fare such as sandwiches, burgers, etc. At this time, the Beach House is open from 9-5 with free shuttle service running between it and the main resort every 20-30 minutes depending on how busy it gets. The shuttle service starts at 10 and ends at 5.
3) You can't rent chairs or umbrellas from the resort, but the HHI Shore Beach Service does, and it's easy to do, though fairly expensive. They set up the chairs and umbrellas for you first thing in the morning. Alternately, a lot of people buy inexpensive ones once they're here (Target,
Walmart, et. al.), use them the entire visit, then leave them for others when they depart. Getting them to the beach isn't that difficult as there is usually room on the shuttle or if you drive (limited parking at the Beach House). You can also bike to the beach with the items if you use a tag-a-long.
4) There are daily activities at the recreation center near the main pool. Most have openings so you can wait to decide which ones you wish to do when you get here. Some fill up fast, so be prepared to sign up immediately upon arrival.
5) Lots of pool activities, too, such as trivia, dancing competitions, and more. You can get crabbing gear from the activities center and head out to the pier, too. So many fun things to do.
Again, the best thing you can do is simply kick back, relax, and soak up the atmosphere. Enjoy!