When I first moved to Madison, I was half a block from Tenney Park - so maybe I'm a little sentimental in my love. Still this is one of my favorite Madison places to plan a run around mostly because it's peaceful and there's always plenty to look at along the river. Tenney's wealth of bridges means I can stretch a run out a minute or two by taking a bit of a zig-zag route without repeating the same path every day. It connects to the Sherman/North Sherman neighborhood on one side of the park and the Atwood/Winnebago area on the other side, so it lines up with favorite biking and running routes.
In the winter, this is my preferred place to ice skate. In fact, this is where I learned to skate last year! Rentals are available and the indoor space is roomy, with plenty of seating. Other than that, nothing fancy. If you're brave and half crazy like me you can ride your bike right up to the shelter to park and skate. There is plenty of ice, so you shouldn't feel too claustrophobic even when there are many kids out.