This might seem counterintuitive, since walking puts stress on your feet. The muscles in your feet are just like any other muscles -- they need exercise to stay strong. You can't really lift weights with your foot muscles, so the best way to exercise them is to just take a walk. It doesn't have to be a long walk, but taking one every day will keep those muscles firm and keep your tendons and ligaments flexible. That can prevent fallen arches later in life. And your dog will thank you for it.
If walking causes sore legs or feet, there are a couple of extra steps you can take to improve the problem. Arch support inserts can keep your feet in the proper position to support your weight during walking. Many stores have special measurement devices that will tell you which type of support is right for your foot. If you've been doing a lot of walking on pavement, your feet might not be up to the impact. Try walking on grass, dirt or a soft cinder track for added cushioning.
Of course, make sure you wear those breathable, well-fitted shoes when you walk.