Larry, I'm not an expert or resident, but I just visited there and here's what I found. If you're downtown, go to 3rd street and take the pedestrian bridge (which looks like a normal bridge at first glance). Once you cross, take a right and ride for a mile or two thru an ugly industrial looking area that makes you think this can't be a bike path. But suddenly you pop into Shelby Bottoms park, and can link to other parks (watch for a bridge) to extend your ride. I managed to burn a couple hours on that route.
The other ride I would suggest is to Percy Warner Park. Go up Broadway til it splits, take the West End split a few more miles to a left on Bel Meade. Ride past all the mansions on Bel Meade until you get to the park, and enjoy a scenic and very hilly ride. It's like a rollercoaster in there, and very pretty.
There's a good shop that helped me out on West End called the Bike Pedlar.