A few suggestions
Well, its a little late for baseball season, but there is always next year. In any event, if you just like the game and don't particularly follow the "stars", I would recommend going to a minor league game. It is fun and affordable, and you get to see people playing baseball largely for love of the game, not for millions of dollars. I was going to recommend the Bridgeport Bluefish, but then remembered that there is another team in the same league that is probably much closer for you -- the Newark Bears.
Other suggestions
The Beach - I could walk on thhe beach all day long, even in winter. In Connecticut, good places to do so are Sherwood Island State Park in Westport and Silver Sands State Park in Milford.
Apple Picking - we usually go to Bishop Orchards in Guilford, but there is also Jones Family Farm in Monroe. I'm sure there are "pick your own" orchards closer to you your home. We often take a picnic -- a loaf of bread, a bottle of wine, some cheddar cheese and a freshly picked apple make a great lunch when eaten out in the orchard under a sunny but cool sky.
Winery visit - we enjoy visiting wineries; you get outdoors and it is interesting. There are a number of vineyards in western CT. Here is a link --
http://www.ctwine.com/western_trail.htm
Vintage train or trolley rides
The Danbury Railroad Museum -
http://www.danbury.org/drm/index.htm
The Shoreline Trolley Museum in Branford -
http://www.bera.org/
The Essex Steam Train -
http://www.essexsteamtrain.com/
On the Essex Steam Train, you can also ride a riverboat up the Connecticut River, which is undeveloped along that stretch and quite beautiful. They also have a dinner train.
Zoo and or Botanical Gardens/Arboretum
I would recommmend the Beardsley Zoo and Bartlett Arboretum in Bridgeport, but for my money a much better trip would be the Bronx Zoo and the New York Botanical Gardens, and it is closer for you.
Interesting small art museums (after you get tired of the Metropolitan Museum, the Frick Collection, and MOMA in NYC)
The Bruce Museum in Greenwich
The Yale Art Museum in New Haven
The Yale Museum of British Art in New Haven
Other attractions in CT
The Norwalk Maritime Center and Aquarium - interesting museum devoted to maritme shipping, shipbuilding etc. Also, South Norwalk, where the museum is located, has a nice section with many little restaurants for lunch.
Things I have enjoyed up the Hudson Valley
West Point - absolutely beautiful location high above the river
Hyde Park - FDR's house is very interesting and they have nice grounds
The Town of Rhinebeck - very nice little village for strolling and window shopping.