The Reefs is a really nice hotel, you are lucky. I've tried to stay there a bunch of times but it is always booked! Make sure you eat at their restaurant (I think it's called Coconuts)--it's tasty.
Some advice: be careful on the scooters! You probably know that you can't rent cars on Bermuda. If you do rent a
scooter, go slowly and carefully. There is a good bus and taxi system if you don't want to rent a scooter.
You are staying in the best parish (Warwick) for beaches, in my opinion. Some of my favorites that are close by, some of them walkable: Warwick Long Bay for waves and lots of pink sand, Jobson's Cove for lots of little fish, Church Bay for snorkeling, and of course Horseshoe Beach for suntanning and swimming (be sure to adhere to the swim flags at Horseshoe--they have rip currents there frequently).
The botanical gardens are worth a visit, as is the perfumery. If you like to hike, there is a trail that used to be a railroad path that you can head out on. Don't go alone though. Also visit the fort at Somerset and the nautical museum if you have the time--they have some cool stuff recovered from shipwrecks of the island. Speaking of wrecks, Bermuda has great scuba diving, if you are licensed.
Food is not the strongest point of a Bermuda trip, but you could try the drinks at the Swizzle in (Swizzle in, stagger out!), and there is a decent outdoor place at the Surf Side Beach Club (the Palms) which is not too far from your hotel. Lobster Pot in Hamilton is pretty good, albeit pricy (what isn't on Bermuda though!) If it is still around, Baileys has good ice cream.
Hopefully this will get you started. Have a great trip!