How long will you have and what time of year? The past three years (and again this year) we've flown into Billings and have spent time in the Beartooth Mountains and Yellowstone. I would highly recommend spending time in both - they are both beautiful and offer different things.
Yellowstone has such a variety of things to see and do - the wildlife, thermal features, hiking - but it comes with crowds. The Beartooths get a lot of people driving through (you should drive the Beartooth Highway - amazing!), but once you get on the trail, you can easily lose the crowds. The advantage of backpacking in the Beartooth wilderness is that you don't have to worry about designated campsites or getting permits like you do in Yellowstone. Cody is also a nice stop - the Buffalo Bill Museum is terrific.
We have several galleries posted that show different hikes/day trips/backpacks in this area:
MT/WY 2011 - Mt Washburn, wildlife, and a lot of geysers in Yellowstone, Bald Peak in Shoshone National Forest, Aero Lakes, Island Lake, fishing, scenery from Beartooths
Beartooths 2010 - Aero Lakes backpack, Republic Peak, Goose Lake, Becker Lake, Iceberg Peak hikes
YNP/Beartooths 2009 - Albino Lake, Martin Lake hikes in Beartooths
If you are looking for lodging, campsites, or permits in Yellowstone - check soon - as they often book up quickly. They get a lot of cancellations though, so you can call frequently to see if you can get something.
Many people combine the Tetons with Yellowstone - but hitting this will depend on how much time you have since you will spend some time at Little Big Horn and you are flying in/out of Billings.
Glacier NP and the Wind River Range (our preferred spot for backpacking) are beautiful as well, but quite a haul from Billings (but again, would depend on how much time you have).
A ton of options - you would have an awesome trip no matter what you choose!