For your trip, wanting to stay at one end of Yellowstone so you can see both parks isn't a bad idea. Grant is at the, mostly, southern end of Yellowstone, but from there you can either go to the Old Faithful area or the lake and canyon areas, you just have to turn left or right, basically. Fishing Bridge is centrally located, on the far east side of the Park, but about midway between the south and north entrances. It's quite a trip from Fishing Bridge to the west side of the park where most of the thermal features are located. If you want to stay in one place and not move, I would suggest this is your best option. And, that location is less than an hour from Grand Teton and Jackson Lake. Grant is a nice place to stay, we've stayed there once and really enjoyed it.
We camp and fish out of Colter Bay in Grand Teton National Park and, sometimes, take a day trip to Yellowstone just to look around. So, coming from Yellowstone to Grand Teton would not really be a stretch, in my opinion. If you did decide to stay in Grand Teton, Colter Bay is our favorite spot, close to Jackson Lake, nice general store, marina and a couple of restaurants. They have about 300 dry camping spots in the campground and a number, I don't know how many, full hook-ups in the RV park but reservations at the RV park can be hard to come by at this late date. And, while not totally centrally located, close enough to do day trips and see all of the park you can see from a vehicle. There are a great number of hiking trails in the Park, but remember to take bear spray as the population of Grizzly's has been getting bigger each year.
