National Trails Day is really all weekend: June 1st and 2nd. Walks include a tour of the Sherwood Forest Zoo, a tree walk through Cedar Hill Cemetery, and more. See the website for all options and details.
Then, just a few days later, if you can handle it, is Connecticut Open House Day on June 8th. This means free or reduced admission, or bonus fun events, at cultural sites around the state. Participating venues in Hartford include: Bushnell Park Carousel, Butler-McCook House & Garden, Connecticut Historical Society, Connecticut’s Old State House, Center Church, Harriet Beecher Stowe Center, Connecticut State Library, Isham-Terry House, Mark Twain House & Museum, and Wadsworth Atheneum Museum of Art. Go to the event website for details.