I would do Beartooth for sure. Yes, there is altitude, had it affected you before? Stay away from caffeine and liquor.
Drink water. Don't do anything super strenuous if you are feeling the effects. Like a long hike would not be good.

I think you need a full day around Old Faithful Geyser area. See the other geysers(which sometimes require an hour or two of waiting). I really liked castle geyser and riverside geyser a lot better than OF. They both went for about 30 minutes, while OF goes for a couple. Plus, we pretty much had them to ourselves. Of course, the the paint pots, springs and other thermal features along the way.
See Canyon, Tower/Roosevelt, Hayden Valley, Lamar Valley. Mammoth is worth seeing, but it is my least favorite area. You will see a lot of elk there though.

Here is a link to our photos from a June 15-20 trip about 4 years ago. We spend 3 nights in Yellowstone and 1 night in Tetons.

http://share.shutterfly.com/action/welcome?sid=0IatW7FqybsWVb

Thanks spitobulldog for your quick response. Great pictures.

Having driven mountain roads in the Sierra Nevada, Cascades, Colorado Rockies and more, I think the 2 most spectacular drives in the west are Beartooth outside of Yellowstone and Going to the Sun in Glacier Nat. Park. Anytime I go to the Yellostone area Beartooth is a must for me.

Thanks boom_boom. How is the drive from Bozeman to Red lodge? Is it scenic? Also how is the drive on bear tooth highway? we are not used to elevation and mountain driving.

If you take I-90 to Laurel and then 212 to Red Lodge you will follow the Yellowstone River most of the way. It is not as scenic as some roads, but for an Interstate it's pretty good. Laurel to Red Lodge is mostly through farm land and not a lot to see.
I don't find the Beartooth as overly hard. There are some switchback sections and some winding, but it is a good, well maintained highway. Just take it easy and don't get in a rush and you'll be fine.

Beartooth is really special. Go for it!

Thanks. We are thinking of 2 options.
1.On June 20th, go directly to Red lodge/Laurel from Bozeman and stay there and early morning do Bear tooth highway and then lamar valley on the way to old faithful lodge.
2.On June 20th, enter YNP through north entrance do Mammoth
and stay in canyon lodge(have blocked it). Then next day drive to Bear tooth highway and then back.
Please give me your suggestions.

I have not done it yet but our family is going to YNP in August. Fly into Bozeman stay one night in Red Lodge. Next day drive down BTH and on to Roosevelt Lodge cabin for two nights. 4th night at OF Inn and last night in West Yellowstone for better access to Bozeman airport. I also looked at lodging in Silver Gate which is at the Northeast ent to Yellowstone. Good luck whatever you decide. Please post a trip report.

I have not heard great things about the Mammoth part of park so I cancelled a reservation at Mammoth Springs and kept calling until I got into Roosevelt and OF Inn.

My choice would be option 1. The Mammoth area isn't as spectacular as other parts of the park and a personal preference, the BTH is best driven FROM Red Lodge rather than from the park into Red Lodge.
AND, just to further confuse you, if there is time, look at a visit to Cody, WY. The Sylvan Pass route from Cody to Yellowstone is also pretty special. But, it all times time.