HHI has its own airport, for smaller planes. Coming from Ohio, if you fly USAir, for example, you would switch in Charlotte to a small plane and fly straight in. Warning: my dh calls these "puddle jumpers"--no jets, just propeller. But the proximity is nice. USAir has the most flights, Continental has a couple, not sure who else flies directly in.
Your second choice is Savannah. After you land you'll have a good 45 minute drive to HH. If you don't do a rental car, there are shuttles that will take you. (but it's really nice to have a rental car while staying there.)
No matter where you fly into, you will probably want to rent a car. The resort is great, but you will miss a lot on the rest of the island if you don't have a car!
I was hoping to get a flight into HH for DH as the rest of us will be driving down earlier in the week. But it was $602 one way! I guess we might not be going now!