Belle Isle is a beautiful park to go to. It is the last thing you would expect to find in the middle of the city of Richmond. You walk across a pedestrian bridge and can follow a trail all the way around the island. Along the way you can stop and hang out on the rocks along the James River, pack a picnic lunch, bring your dog (on a leash, of course!), and maybe even take a swim in the river if the water isn't too high. There are also some bike trails, although I have never used them personally.
It can be a little crowded on a summer weekend, but it is a beautiful (and free!) way to spend a day!