I went to Yellowstone for the first time last month.
The only peak I hiked was Mt. Washburn. It was an easy walk up a fire road but the view from the top was unlike anything I have ever seen. There is a fire look out on top and there is unobstructed view in all directions.
I really wanted to do Electric Peak, but was uncomfortable heading out on a 20 mile hike solo in unfamiliar territory and I couldn't convince anyone in my group to join me. Lucky I was able to convince my wife to hike the 21 mile Mary Mountain trail. The trail is basically flat (the only reason my wife went) and crosses through the central plateau. We got dropped off at the eastern trail head (near Canyon) and then picked up 9.5 hours later on the west side(near lower geyser basin). This hike wasn't very physically demanding, but mentally it was taxing! There are buffalo everywhere that you have to avoid, the trail is faint to non existent, and we were on constant alert for bears. We were off trail for probably 5 of the 21 miles because of all the game trails and most of the markers have been knocked over by buffalo. Lucky I had my GPS and plenty of way points to stay heading in the right direction. We did come across 2 grizzly grazing in a high meadow. I think they were eating bugs. They were probably 200 yards away, but definitely made my heart race. At about half way we had lunch at Mary Lake. Very beautiful lake and not a soul there. In fact we didn't see any people the entire hike. Had a scary few minutes with a pissed off buffalo on the way out, but we managed to get around him by leaving the trail and heading off into the woods.