Hike it Baby is an organization dedicated to “getting families together and out into nature with birth to school age kids.” http://www.hikeitbaby.com
AllTrails is an app that will show you nature trails in your area with maps and an idea about how difficult a given hike (walk) might be.
The New Haven Land Trust runs educational programs, supports a couple of community gardens, and manages several interesting land preserves in the New Haven area. http://www.newhavenlandtrust.org/
The best way to cultivate a close affinity to the natural world is to repeatedly visit those natural places that are close to you and easy to get to. Make sure you don’t overlook your local urban parks! Check out the New Haven Parks Department webpage and their face book page for special events and programming. http://www.newhavenct.gov/gov/depts/parks/default.htm
Here are some of the trail hikes in the New Haven area. http://www.ct.gov/deep/cwp/view.asp?a=2716&q=325276
Don’t overlook the Farmington Canal Trail that starts in downtown New Haven and goes into Hamden and beyond! http://www.farmingtoncanal.org/Maps.htm
The Sleeping Giant State Park Association http://www.sgpa.org/
Great blog about West Rock Ridge State Park http://westrocktrails.blogspot.com/p/west-rock-user-guide.html