Yellowstone is a big place to try and see but staying at Bridge Bay might be pretty good for you. You're right on Yellowstone Lake on the way to Canyon and up to Mammoth, over to Norris and down to Old Faithful. Usually, most people travel one loop at a time. On your trips around the Park, you will find many opportunities to hike, if that's what you want to do but there will also be numerous "animal jams" for bear, moose, bison or whatever other animals choose to show themselves to the tourists. We fix a lot so we go to the Colter Bay campground and fish and when we get a little tired of that we will jump in the truck early and take a little trip to Yellowstone, about 45 miles to the south entrance. Usually, we can see one loop fairly quickly on each trip. We don't hike so we aren't looking for hiking trails.
As for Grand Teton, I don't agree with the folks who say Gros Ventre is closer to the action than Colter Bay Campground. Colter Bay Campground has some electric hookups that were built a few years ago but are partially reserved for folks with disabilities or access passes. So, it's dry camping but they have two place to fill with water and dump waste so it's not a big deal. Cost is $24 per night compared to the RV park at abut $58 per night. If you have the Senior Pass, the campground is $12 per night, a wonderful bargain. Colter Bay sits right on Jackson Lake and has a marina which offers cruises throughout the day, including lunch and dinner cruises to Elk Island, in the shadow, almost, of Mt. Moran. There are some hiking trails around Jackson to Swan Lake and Hermitage Point, you can then drive to Signal Mountain and over to Jenny Lake, about 15 miles away from Colter Bay. Get on a boat at the visitor center, go over to the Hidden Falls landing and hike to Hidden Falls and Inspiration Point. After that, you can drop over to the Moose entrance and take the Moose/Wilson road and go to the LSR Reserve, the newest part of Grant Teton and hike up to Phelps Lake where the Rockefellers had their compound prior to donating it to the Park. They have done such a great job of taking it back to nature you can't even tell there were almost 30 cabins and residences there. After you have done that, it's a fairly short drive over to Gros Ventre where one of the biggest landslides in US history occurred, making Slide Lake about 15 miles off the main road to Jackson. Then, one night, you can go into Jackson and take in the Bar J Wranglers' show with a chuckwagon dinner and wonderful western entertainment with great music and comedy. And, if you really want to be in a zoo, just take your trip into Jackson a little early and see all the tourists, of which you will be one.
As you can tell, I really like Grand Teton and much prefer Colter Bay to Gros Ventre, and yes, I have stayed at Gros Ventre Campground once when I couldn't get into Colter Bay due to construction last year. It certainly wasn't "close to the action" for us to go fishing. Took about 45 minutes to get to a launch area from Gros Ventre compared to two or three minutes at Colter Bay. If you don't fish, not as big a deal.
John A. Lichty