The Old Market / Heartland Park / Central Park Mall is a area to walk around. You can also rent bikes to ride around the area.

You should also make some time to visit the Zoo.

Here's a good site to find more info on Omaha - http://visitomaha.com/

Specific to hiking and trails, this could help - http://visitomaha.com/vandr/Sports.a...EC&sn=visitors

And yes, there's a Whole Foods:
http://www.wholefoodsmarket.com/stores/list_NE.html