Walking, biking, rollerblading, running, fishing, etc.! Your hotel may offer bikes or you can rent here (just minutes away from the hotels): https://palmbeachbicycle.com/
The Flagler Museum
1 Whitehall Way, Palm Beach, FL 33480, (561) 655-2833
Whitehall is a 75-room, 100,000 square foot Gilded Age mansion! Right on the island and beautiful!