Nature is everywhere -- we just need to learn to see it
Go outside! Make a commitment to spend a certain number of minutes outside every week. Ideally, make at least some of these minutes about exploring and noticing, rather than exercising. If you have kids, get them involved, too. A great initial project would be to try to identify the species of every tree on your block. Or hang a bird feeder and keep a list of species that come by. Or find a local piece of "wasteland" and take some photos of the plants and animals there. (Always be careful — some fantastic urban wildernesses also feature sharp broken glass, rusty wire and the like.)
Plan a weekend trip to non-park nature. Identify a public area within a day's drive where the crowds are thinner and the opportunities for interactive experience greater. In the US, start by contacting your local state fish and game office, forestry service office, or BLM office to learn more about fishing, hunting, swimming holes, mushroom picking, rock or pine cone collecting, and in some places, even making a bonfire on the beach!