The whole island is bikeable with trails. We would recommend exploring Sea Pines by bike (most hotels/airbnbs come with bikes and there are several bike rental spots within Sea Pines) and all the beautiful streets it has to offer.
Hilton Head has 13 miles of continuous beach. Within Sea Pines there is an entrance to the beach every 150 feet along the roads on the water. The main beach entrance with parking is at the Beach Club within Sea Pines or Coligny Beach. Bikes can be ridden on the beach. Sunsets are beautiful.
Lots of tennis courts on Hilton Head. Smith Stearns Tennis Academy (run by Stan Smith former Wimbledon Champion) is one of the best in the US and offers lessons for reasonable prices
In total there are 30+ golf courses in Hilton Head. There are 3 within Sea Pines (Atlantic Dunes, Heron Point and the famous Harbourtown which hosts an annual PGA event)
Located at the southernmost tip of the island on the Calabogue sound with a great view of the water. The Salty Dog is a famous spot to enjoy breakfast, lunch or dinner with live music daily. You’ll see a lot of the people on the island wearing Salty Dog t-shirts. Reservations not necessary
Located at the north end of the island, this is one of the best places to enjoy a nice sunset. There are 3 restaurants (2 are walk-in only while you can make a reservation at Skullcreek Dockside) and a ton of games on the restaurant grounds including ping pong, foosball, etc