There are plenty of choices to run/hike/walk in and around Chattanooga but here are 3 very nice choices.
Downtown Chattanooga Riverwalk
The Chattanooga Riverwalk runs along the Tennessee River from downtown all the way to the Chickamauga Dam near Highway 153. It is great for runners and walkers of every fitness level, and while bikes can be there too, runners and walkers do have the right of way. It is an almost ideal place for people with jog strollers. Total length about 9 miles.
Easily reachable on foot from downtown hotels. Can be started right behind the Aquarium or right at the sculture garden in the Bluffview Art District. If you drive there, park near the 200 block of High Street.
Lookout Mountain Guild-Hardy Trail
The Lookout Mountain Guild-Hardy Trail is a walking and biking trail located on the side of scenic Lookout Mountain and provides access to a multitude of trails on Lookout Mountain, the National Park Service land and Reflection Riding. The GH Trail winds through forested mountain slopes littered with historical ruins, Ruby Falls, and the Cravens House.
Directions: From Downtown Chattanooga, take I-24 to Lookout Mountain/Broad Street exit. Take Broad Street to Tennessee Avenue and make a right turn onto Ochs Highway after The Incline. The Ochs Highway trailhead parking lot will be on the left.
Walk or run the Bridges
Fom any of the downtown hotels it is quite easy to do a nice walk or short run of the downtown bridges. Go up either 1st Street or 2nd Street until you hit Walnut Street, turn left and go across the stunning pedestrian Walnut Street bridge. Turn right onto Frazier Ave until you get to the Veterans Bridge and then turn right on it. Veterans Bridge eventually turns into Georgia Ave and you'll stay on it ontil you get to 4th Street. Turn right on 4th and you'll get back downtown.
More choices for running and hiking are listed on the http://www.outdoorchattanooga.com