i copied a walking tour from the national geographic website and i did it today. basically you walk around the island starting from the southern tip at the clarion hotel where the pedestrian walkway starts and you end up at slussen. i think it took a couple of hours, but i went to see the langholmen prison (now hotel/hostel) and stopped to take a lot of pictures. i also went up the katarinahissen elevator (10kr) in slussen so i could take pictures from the gondolen.
the tour is great because you are really always at the water's edge. here's basically the starting point:
here is the floating bathhouse from 1930, the last one of it's kind