I'd take you mushroom hunting (but really just exploring and running around in the forest this time of year), maybe pick up a trout on the way home to grill. That's a few days a week for me (the trout leas though).

As for touristy things here in Zurich - it's not really a tourist city. When we have guests from overseas we do have a set of activities to bring them on. When I've offered to bring them in the forest to find mushrooms/berries/etc they're usually not so interested.