Wendy covered the most popular destinations. There are a couple I recommend. First, visit
Riverside State Park. It's on the northwest side of town. The Spokane River flows strongly through the 'Bowl and Pitcher' area. Walk over the swinging bridge for a mild thrill. Make it more thrilling by getting the bridge to swing!
Second, drive up to the
Indian Rock paintings, also in North Spokane. A quick walk from the parking area will lead you to the paintings (they are behind metal bars to keep out vandals), and if you like you can hike the trail further down the Little Spokane River. It's a gorgeous area that's off the beaten path and not many people are aware of.