I used to live up in that area. Welcome to CT!
Let's see. UConn has the dairy bar (great ice cream!), and if things are the way they used to be, you can walk all over Horsebarn Hill and see the different animals they raise there. In the spring, they used to let families into the barns to see the babies being born. It's also a great place to go at night to see the stars and any astronomical event that might be going on (Mars being visible, meteor shower, eclipse, etc.).
If you go down to Willimantic, there's the Frog Bridge, the Willimantic Food Co-Op, and various events at Eastern University. There also used to be a boombox parade every Thursday during the summer, with sidewalk markets and the like.
If you drive down to Niantic, off Rte. 395, there's a children's museum with plenty for little ones to do, and the Book Barn, which is an awesome used book store on an old farm. They have rescued dogs and cats, and a set of goats for kids to feed and pet. It's run by a very friendly family. Plus you can get a ton of great kids books for $1 each.