Hi, we'll also be in HH next month. Are you renting a car for the week? Here is what we like to do...
Day trip to Savannah- park at the lot at the visitor's center and pay to take the trolly tour. You learn a great deal about the city, can depart the trolly for lunch/shopping/etc, and return to it. The tours give not only historical information, but pop culture information, such as where movie scenes were filmed.
Day trip to Charleston- take a tour of the city in one of the horse or mule drawn carriages. This is another great way to get an overall view of the city and a brief history lesson. Don't miss Rainbow Row and the waterfront.
HH- we stay in Sea Pines and love to take advantage of all of the biking trails. We literally bike about 20-25 miles per day while we are there. Go to the top of the HH lighthouse and take in the view, walk the beach, relax. We also explored the Stony Baynard Ruins, biked out to the Salty Dog, and explored the Canal Trail. If you are into birding, take binoculars. The variety is amazing. We also went to the stables in Sea Pines and went for a trail ride. There is also a weekly hayride.
We have a little one (he just turned 2, enough said), so we don't eat out much while visiting. We did manage to go to Wild Wings Cafe last time we were there, and if you like wings and a good beer, it was a good stop. Definitely not fine dining though.
I hope that gives you some ideas. Tennis and golf are also available.
Jeni