First, I assume you have reservations or are going before the weekend starts. CG fills up real fast. My favorite campground there ( when I stay in a campground ) Is Sheep Creek, space 27. A good hike in the NF is up Deer Cove trail. Just a bit past Grizzly Falls ( a good place to pic nic ) but before you re-enter the NP at Cedar Grove. Both the trail and falls are on your left as you head toward CG. It's fairly steep for the first 1-2 miles and there may be no water for about 4 miles. Depends on how dry it is. The trail crosses a couple small streams. Check with the rangers at the CG station. They'd be able to tell you. Either way plan on carrying plenty of water as it will be very hot on the trail. But once you reach the top you can down into the canyon, Lookout peak across the way ( some cool places to camp there but no "official" camp sites [:D], and on a good day you can see all the way to the San Joaquin Valley. It is ok to take your dog for a walk on the road of course, which is relatively levelfrom the campgrounds to roads end. From the campgrounds you can walk to the roads end. The road crosses the river several times and Roaring Falls is a don't miss spot along the way. Check with the rangers about taking your dogs on the unguided nature trail, which is really a dirt road that runs almost from roads end, along the river, and comes out behind the stables just across the river from the campgrounds.
Another good place to hike and take the dogs is the Jennie Lake area. You have to go all the way back up to the top, take 180 where it forks ( you'll see it when you go there ) and go south. Look for the signs that say Buck Meadows / Jennie Lake. There are actually two lakes, small, and the are only a couple miles in on a easy to moderate trail. Lots of people back pack with their dogs there. Makes for a good day trip.
Feel free to e-mail me direct with any questions.