Great places to get out in the Great Outdoors: every state has at least one National Wildlife Refuge, and most major cities have one within an hour’s drive.
There are well over 500 National Wildlife Refuges, and that number can continue to grow. You can find National Wildlife Refuges by region, or search by state.
Many National Wildlife Refuges offer fishing, and early June is a fine time to take kids on a fishing trip, during Fishing/Boating Week: most states waive fishing license requirements a day or two, and some places have a special "Take A Kid Fishing Day", with hot dogs, prices, certificates, stocked ponds, etc.
For example, in Maine, the Aroostook National Wildlife Refuge has an annual Children's Fishing Derby, with a pond stocked with trout; Trinity River National Wildlife Refuge hosts an annual "Free Family Fishing Day"; in New York State, Iroquois National Wildlife Refuge has an nnual Youth Fishing Derby for ages 17 and under.
Other events continue through the summer: Lostwood National Wildlife Refuge (in North Dakota) holds a Birding Festival in June, including guided tours and a Prairie Wildflowers Walk; in New Jersey, Friends of Great Swamp National Wildlife Refuge (New Jersey) has special days, such as a day dedicated to butterflies, or dragonflies, with guided walks.
Through the year: Check the National Wildlife Refuges site for fall events, such as butterfly hunts, fall festivals, or collecting seeds for National Public Lands Day, and for events through winter and spring as well.