I think you might get more specific info if you put your city in the title of your post.
However, in general, I think the public library is a huge untapped resource in most communities. They often sponsor musicians and storytellers as well as craft projects, sometimes around a theme (e.g. environment for Earth Day). They also usually have a bulletin board for flyers, and the librarian is a very knowledgeable person. Our library has a ton of free or discount passes for children's museums, the science museums, and more.
Town recreation departments sponsor things like July 4th and "Hot Summer Nights" block parties, community bands exist, and a lot of senior citizen complexes have summer band concerts for free (in our area, people bring lawn chairs and enjoy the free music, and sometimes there is a sundae bar). A lot of our local garden centers have small petting zoos and sell food (so people don't feed junk food to animals). Sometimes they have a planting day in the spring to help families plant seeds or learn to make a vegetable garden. (An added benefit is that kids tend to eat what they grow!)
State and national parks are great places for hikes and nature walks and picnics, sometimes canoeing and kayaking. The Audubon Society and other land trust organizations often are trustees for open space, and many have self-guided or naturalist-guided tours.
Many organizations and museums have links on their websites that connect you with similar organizations. A membership in one museum in our area often gives free admission to other museums in the network, all within an hour's drive.
In our town, the fire house welcomes kids to come and learn about the department - one of the big problems in fires is that kids hide, so firefighters often gear up to teach kids to run TOWARD them. Fire stations are either very quiet or chaotically busy, so unless there's an emergency, they often allow kids in. Yours might be a little old to climb up in the driver's seat of the engine, but learning about the types of equipment or the inside of an ambulance can be educational and also a great experience if there's ever an emergency down the road.
Community Supported Agriculture is a big movement, and our local farms have all kinds of opportunities for pick-your-own fruits/veggies, whatever's in season. We're in the middle of berry season, and in the fall it's apples and pumpkins. A lot of the farms have hayrides, petting zoos or fresh ice cream from their own cows.
I think it's great that you get out there with your kid! I always did with my son, and he really blossomed from not being in front of a TV or computer screen the whole time.