I assume you mean Gardiner? I'm in SW Montana, not too awfully far from Gardiner as the crow flies. I'm always eager to give suggestions - it's one of my favorite topics. lol. How long will you be there and for how long? I actually like your plan of just "enjoying." I think too many folks try to see "everything" and end up not really SEEING anything.
If you are going to be in Gardiner, I would consider a drive on the Beartooth Highway an absolute must-do. Enter the park right there in Gardiner, go through Mammoth Hot Springs and head toward Tower Junction and, eventually, the northeast gate of the park near Cooke City, MT. In my opinion, the northeast corner of the park is the most scenic area. You'll usually find the largest bison herds in the park in there along the Lamar River valley and the Soda Butte Creek area. It's a great place to just stop for 15 minutes or so to just "take it in."
After leaving the park at the northeast gate, you'll hit the little one-horse town of Cooke City, MT which is the unofficial beginning of the scenic Beartooth Highway. From Cooke City, it's about 100 miles to Red Lodge, MT, but it may be the most incredible 100 mile drive in North America. The first time I drove it, I wanted to stop about every 1/4 mile to take a photo. Red Lodge has a cool main street that is pretty vibrant with shops, restaurants, and bars. The round trip is mostly an all-day thing, but it's well worth it.
If you are interested in a really scenic hike in Yellowstone that isn't very strenuous, you will pass a great trailhead on your way east through the park. Soon after you make the turn east after Tower Junction, you'll pass the Yellowstone River Picnic Area on your right. It's a small loop with maybe 20 picnic tables, but the trailhead starts right there. The trail goes up a little hill, but after the first 1/4 mile or so, it levels out at the top of the canyon and parallels the river. You can decide how far you want to walk before turning back, but it offers some pretty spectacular views of the canyon for a relatively low investment. You'll probably see some other people on the trial, but this is certainly one of Yellowstone's hidden gems that 99% of the tourist drive right by. Just one note: don't be a jerk by parking in one of the picnic sites while you walk this trail. Save those for actual picnickers and park along the road just outside the picnic loop.
Other than that, just venture wherever the wind blows you in the Gardiner area. If you see a trailhead that looks interesting, just stop and try it out. Don't get too caught up in "checking the boxes" of the famous sights. There's a reason they are famous, but every other tourist in America will be there checking the same box and I find it more enjoyable to discover less crowded attractions that are nearly as impressive (and sometimes, even more).
