Malmö is not a massively big city. The old town with a few parks around is definitely walkable, but Malmö is not just the old town. After living here for a while, I can safely say that the best way to experience the city is on a bike. You can cover a lot more ground that way and see places that are not that easily accessible by bus. On top of that, Malmö has a great network of cycling paths, so you get from one point to another in no time.
The bike also gives you freedom to venture a bit out, maybe along the beach to Klagshamn or up north to Lomma. In any case, you will be sightseeing and exercising at the same time. If you don’t have your own bike, you can always rent one from a number of bike-share platforms that operate around town. You only need to remember that in some parks it’s not allowed to cycle and for your own safety don’t forget to wear a helmet – at the end of the day, the incredible Hövding was invented not far from here, in Lund.