I spent every other summer there between birth until my daughter was born six years ago. My great grandfather built a ranch in Moose, WY (near the Grand Teton National Park entrance) in the '50s, but sold it to the park service before he died (with a lease on it that expired five years ago). It is an amazing place. You are so lucky!
Since I haven't been there since 2004, I am sure restaurants have changed somewhat, but one that I know is still there is Dornan's Chuckwagon in Moose. If you are planning to spend time in the Grand Teton National Park (GTNP), then you'll drive by Dornan's (if you use the South entrance to the park). Dornan's is a family-owned/run business with a grocery store, gift shop, Chuckwagon restaurant, liquor store/wine store, bar and pizza/pasta restaurant. I used to work there during the summer in high school and college. I LOVE Dornan's breakfast at the Chuckwagon on a clear morning. The view is beautiful.
There was a restaurant in Jackson called Sweetwater Deli/Restaurant. We used to eat there for lunch and occasionally for dinner. Lunch was MUCH better... I would skip dinner there, but lunch was very good.
Nachos and margaritas (if you drink) at Signal Mountain Lodge in the GTNP are sort of a tradition for many locals and summer employees. Also a beautiful view there!
There used to be a tram to the top of Rendez-vous Mountain in Teton Village. It shut down, but I heard that someone re-opened it. If it's open, you should take the tram to the top of the mountain. We have lots of family photos from the top!
A scenic float trip down the Snake River - do early AM so you can have a better chance of seeing wildlife. If they do meals, the breakfast used to be good.
The Moose Ponds on the Moose-Wilson Road are great for spotting moose. Early morning is usually best.
Death Canyon Trail Head - there is an old dude ranch (White Grass) and a dirt road that you can walk, but can't drive. If you walk the road just before sundown, you should see lots of elk. That is where my family's ranch was and we'd have nightly visits by herds of elk. Be sure to take a flashlight to get back to your car.
If you like to hike, there are lots of great hiking trails. An easy one is to Hidden Falls. You have to take a boat on Jenny Lake or walk around the lake. The Falls are pretty. If you're up for a longer hike, get lots of water and go on to Lake Solitude. It's a long hike, so you have to be prepared to be gone all day. There is also horseback riding around the lake.
I could go on and on about things to do there, but these are some highlights. Have a great trip!!