So I am talking to my son as I type and he expresses the following.
#1 Hike up Mt. Washburn near Mammoth Hot Springs. He said its a relatively easy hike, but you should wait until your acclimatized to do it because you'll be hiking up to about 10,000 ft. Of course, like Jan said, you'll be starting at nearly 8,000 ft. It would probably take you at least a day to acclimatize if you are living near sea level like we do.
#2 Go to Boiling River, also near Mammoth Hot Springs in the northwest part of the park. He suggests getting there early in the morning to both catch the sunrise and beat the crowds.
#3 Visit the terraces - again near Mammoth Hot Springs (since there is so much to do in this area he says you might want to look at the hotel in Mammoth Hot Springs - no idea what the prices are though. Gardiner is close to Mammoth Hot Springs too and might have more reasonable .
#4 Visit Yellowstone Lake
#5 Have to see Grand Canyon of Yellowstone in the middle/south part of the park.
#6 Madison has a great visitor center that's just been remodeled.
# 7 Mud Volcano or any other geyser basin - Often you'll find things a lot more interesting in the less well known geysers.
What to avoid:
#1 Old Faithful - I know I know - how do you go to Yellowstone and avoid Old Faithful. He says its not worth it and you'll be waiting in a line like Disneyland and probably not catch much of the eruption because it would likely be concealed by a baseball cap or two or twenty or kids on their parents shoulders.
# 2 Bechler - He says its beautiful but FULL of mosquitoes and horseflies.
# 3 Bison - They're aggressive and faster than you and too many people think they are just like cows and get too close. BTW - they can jump so don't think a small fence would protect you. Stay clear and enjoy from afar. He says if you encounter one stay at a 45 degree angle to him or he will think you're challenging him (get out your protractor)
#4 Grizzlies - need I say more - also enjoy from afar far far
Things to be aware of or think about
#1 This is from me as a person who lives low and has climbed to heights above 10,000 ft at least a few time in my life - Drink lots of water and bring plenty of snacks while out and about even if the temperature is cold or just nice. As a person coming from a low area to a high elevation your body and the bodies of your family will need more water and food than usual. One funny way that altitude affects people is it tends to make them feel like they are not thirsty or hungry when their body most needs to be fueled. That results in headaches and crankiness, which will really ruin your day. At worse it could be dangerous if you exert yourself too much or too quickly. HAPE has actually been known to occur among skiers who come from a low area and fly into the ski resort and immediately hit the slopes trying to get the most out of a short vacation. Its just something most people don't really think about. I've NEVER developed HAPE though but I have gotten cranky and headachy due to altitude adjustment. Made me very whiny and my poor BF fed up.
# 2 My son says he can't emphasize enough to bring clothes for every type of weather. He said while he was working there last summer he got sunshine, rain, hail, snow and sunshine again all in one day.
Sorry for such a long post! My son loves the area and can't say enough about it. He has much more to say, but we gotta quit somewhere
I hope we haven't inundated you.