It goes without saying that heading into the city is definitely something to do. If you're staying at the Hyatt Regency Greenwich, you are SO close to the Old Greenwich Metro-North Railroad station. You can hop on a train there and go right into Grand Central Station for about $15. Even if you plan to rent a car while you're here, we would still recommend you take the train into the city as parking can be difficult.
Westchester Mall is not far and has great shopping and dining.
Greenwich Avenue is "main street" in downtown Greenwich. It's a historic street with unique shops and dining.