Across Jenny Lake is Hidden Falls & Inspiration Point. You can hike around, or take a ferry. The day we were there it was 30 degrees and snowing. We went on a Ranger Led hike from the Visitor Center across the lake on the ferry, up to Hidden Falls, and then climbing up to Inspiration Point.
The Ranger, Laurel, explained the history of the park, and how it was formed on the 90 minute hike which was very worth it.
We hiked about 20 miles in Grand Teton National park. Most of them were long hikes, but none were too strenuous. We got caught in a thunderstorm at Taggart Lake, so we only did part of that trail, and we cheated and took the shuttle boat across Jenny Lake for Inspiration Point.
The famous abandoned Barn of The Tetons lies on Mormon Row, a historic district of Grand Teton National Park. When we were there there was a family of Coyotes living in the first building, as well as antelope, and bison in the fields. The sky didn't cooperate, it was stormy overcast, so no iconic photos of the barn, next time it seems.