Much of Fairfield county (which incorporates Greenwich and surrounding towns) is on the coast. There are many parks and beaches to explore with fantastic views of Long Island Sound. Depending on the season, it's a great place to rent a charter fishing boat with buddies or go sailing with the family.
Not all beaches have the same rules. Check rules on pets, containers, hours, etc. before you venture out.
The Maritime Aquarium in Norwalk is a must-visit if you have kids.
Greenwich and the neighboring Stamford are known for being business (financial) meccas given their close proximity to New York City. Stamford has a great collection of mid/upper tier hotels that cater to business professionals/travelers. If you want some "New England feel" to your lodging, there are a great collection of inns off I-95 between Cos Cob and Greenwich.
Take It or Leave It:
Greenwich is 38 minutes from Grand Central Terminal by train, so it's a real alternative to staying in Manhattan.
If you want to do the "inn" thing, check out the Homestead Inn.
Dining in Greenwich gives you many upscale, fancy choices such as Valbella and Barcelona. Old Greenwich restaurants have more of a homey, quiet feel. Stamford has a great collection of Italian restaurants, and the downtown area has many of the staple national chains. Head to Norwalk for pizza, deli sandwiches and fried seafood.
Some Greenwich restaurants spend more time on their decor than their food. Check reviews, and don't judge a book by its cover.
Sports fans should check out Bobby V's in Stamford.
Greenwich and neighboring Stamford both have a good amount of bars that cater to the after-work, happy hour crowds. People who plan to tie one on usually head to Stamford for late night shenanigans, live music and dancing. The area around Wall Street in Norwalk has some great dance clubs as well.
Many locals/visitors tend to go into the city (New York) for weekend night partying.
Check out "Rhythm" in Norwalk for great live music.
Stamford and Greenwich do their best to keep up with the shopping prowess of nearby New York City. Stamford Town Center has stalwarts Macy's and Saks to help draw upscale shoppers. Nearby Post Road in Westport offers every store the female mind could dream up, including an amazing Anthropologie store. On the way there, you can stop in Norwalk at the Town Square Shopping Center to pick up basics and find hidden gems.
Westport has the best shopping out of any neighborhood in the area. It will get crowded on Sundays, so try and shop mid-week.
Best Local Souvenir:
A lobster, preferably real but stuffed will do.