Cycling
Cycling is an excellent way for children to see and experience the world around them, and with the correct equipment it's very safe. It is also very healthy and doesn't put pressure on children's knee joints. You can take them for a ride in your local path, around the block or along a beach. As they get older, you become more comfortable with their riding and they get know the area well, you can allow them to go cycling on their own.
Birdwatching
If you're looking for a slightly more academic or scientific activity for children, you can take them birdwatching. It can be done in both rural and urban areas, and provides them with valuable knowledge their peers are unlikely to have. You'll need to buy them a book to help them identify the birds they see and stay with them for the first couple of hours to help them get going. You can make a game out of it, and get them to tick birds off in their book as they see them. If they've got a camera they can try and some photos of the birds they see.
Swingball
If you don't have the space to play badminton or access to a tennis court, swingball is a good alternative. It requires minimal space, can be played with either one or two players and can be stored indoors once the game is over. It requires a post that is held up with a plastic base filled with water or sand and that has a piece of string with a tennis ball attached. Players hit the ball before it completes three successive turns of the post.
Hopscotch
Children can play hopscotch alone or with adult supervision. You can buy sets from stores, but it's easier to draw out a court on concrete with a piece of chalk.
Vegetable Growing
A long-term activity is vegetable growing. It provides children with an appreciation of where their food comes from and a sense or reward and satisfaction when the plants grow. If you've got a large garden you can free up a small space or alternatively you can hire a plot at a nearby allotment.