For hiking, there are of course several easy/moderate hikes in the northern part of Yellowstone, which you can learn about through the NP resources or All Trails. In Paradise Valley, the most well-known hike is Pine Creek Falls and Lake. The falls are an easy, fairly flat 1 mile hike. The trail goes another 3 miles up to a spectacular alpine lake in a glacial cirque. It is an ambitious day hike (3,500' elevation gain), but many do it, and the prize at the top is phenomenal. If you're into waterfalls, there are several in Hyalite Canyon, which is also a great area to hike, but it does require an hour drive around through Bozeman. Wish I could give you good recs around GT, but it's been a while since I've hiked down there. If you're not familiar with hiking in this area, please have a can of bear spray on you and available...grizzlies are prolific in some of these areas, and they can be aggressive if they feel threatened.
If you want a scenic drive that would take most of a day, the Beartooth Highway is awesome. The Beartooths and Tetons are the prettiest mountains in the area, but only in the Beartooths can you drive up and over a high mountain pass. The views are spectacular. From Sage I would recommend driving a big loop down through the park, hopefully see some wildlife as you drive through Lamar Valley, exit the park at Cooke City and the pass will take you over to Red Lodge, a cool little town. Then come back through Livingston. Could also do it in the reverse direction. This makes for a great day.
Regarding food, I have no idea how any of these are currently affected, but I'll give some recs and you can look into them. Of course you have the restaurant at Sage, which is good. You're also very close to Chico Hot Springs, which has both upscale and low-key options. My favorite in the area is the 2nd Street Bistro in Livingston, which Anthony Bourdain visited in one of his episodes. There are also a number of other good place to eat in Livingston if you want to drive there, Gil's Goods being another favorite. The saloon right there at Emigrant also has good food. Pine Creek Lodge is another fun place to check out for dinner or weekend brunch. There are some places in Gardiner, but I haven't had much there that totally blew my socks off. Definitely check on dining availability in the park before you go, as you might want to throw lunch in an ice chest before you go in. Finally, the Wildflour in Emigrant is where it's at for your morning fix. Awesome cinnamon rolls and other baked goods, and good coffee.
The only other specific recommendation would be to go visit Chico Hot Springs, which will be right around the corner from you. The hotel is 100+ years old, and the pool is usually a fun mix of locals and visitors. They have bar service at the pool, so it makes for a great way to go spend an hour or two in the evening.