Wow. Those are some young mountains. Plus bears. We saw quite a few bears in Grand Teton compared to none in Yellowstone. People we spoke to often had the exact opposite experience.
The nearest nice town is Jackson Hole and it really is a nice town, but a bit too isolated for us. So most of our time was spent exploring the national park and only a bit of time was spent in the town of Jackson Hole itself. We camped in Signal Mountain Campground on Jackson Lake and spent some time kayaking, paddleboarding and generally exploring the park.
Signal Mountain Campground is first-come, first-served. So we got up very early from our Sinks Canyon State Park location in Lander, Wyoming and tried to drive fast-ish with the trailer the 130 miles or so to get a spot at Signal Mountain before everything filled up. Each time someone with camping gear on their roof passed us, we worried that they were taking our spot. But we made it and got a decent spot. There are only a few trailer-sized spots right on the lake, but we were not lucky enough to get one of those.
We took our kayak out to one of the islands in Jackson Lake, enjoyed the peace of the evening, and then paddled like heck back
to get to the mainland before darkness fell entirely.
A lot of people with trailers or RVs recommended Colter Bay Campground, which is further up the lake. But we thought it was crowded and not as nice. They did have full hooks-ups, though...
We'd certainly come back if we were in the area. For example, if we wanted to go to Yellowstone again, we would definitely include a few days at Grand Teton.
You aren't allowed to live in a national park, silly! (Actually, some people are grandfathered in, but neither of us are grandfathers.)