When you start from Bozeman you could stay in Gardiner at the entrance of the park.They have a motel 6 .I would suggest three days to visit the park.Then on to the grand Tetons.In Jackson there's also a motel 6.For Salt lake city 2 days is enough.If you want more time in Yellowstone you could visit the Beartooth mountains,Cody with the three great museums in one.Both destinations are close to the West entrances of the park.Paul

If a flight to Bozeman is expensive from LAX, think about flying to Portland or Seattle and taking the Empire Builder train overnight to Whitefish Montana. This way you can also visit Glacier National Park as well as Yellowstone.
I don't know what your deal is with a one way car rental.
With enough time you could also visit one of the Southern Utah National Parks flying back to LAX from either Salt Lake City or Las Vegas.

Actually, both Cody and the Beartooth are on the East side of Yellowstone. Cody is out the East entrance to the Park and the Beartooth is the Northeast.

We were at Yellowstone and the Grand Tetons in early to mid-September 2011. Still lots of animals to see. The elk were everywhere at Mammoth, mostly females and young males as it was still pretty warm and the big males hadn't come down from the higher elevations yet to begin gathering their "harems".

Some people don't care for the Mammoth area as much because the terraces aren't as magnificent as they were many years ago, but it's a handy base for a few days to get to some of the wildlife viewing areas and they do have cabins. Our first trip to YS years ago was in the summer and we saw tons of children enjoying the large grassy area by the cabins. A place to let them burn up some energy was always a consideration for me when we traveled with our children. You're also handy to Gardiner just outside the Park to the North, for a place to eat out or pick up some additional snacks and such. Also, we went out to the terraces at night and listened to the coyotes howling. Your kids might like that.

The bison are easy to find. We've always seem the most in the Hayden and Lamar Valley areas but you might run into them almost anywhere. Along the road between the Northeast entrance and Tower is the only place I've seen a wolf. One as it crossed the road, the others were tiny dots in the distance. Sightings of coyotes are quite common. We've seen a lot of mule deer along the west side geyser basins. The lone big male elk we saw last September was in the Madison Valley. Other times we've seen them along the road between Mammoth and the NE entrance. Have yet to see a bear in YS but others have at the times we were there - we just missed them.

We've seen moose every time we've been in the Tetons, never in the same place twice.

Your best times for seeing wildlife are early morning and late afternoon, but with a little one, getting up and out really early might be hard to do. We always were up before so dawn, explored around before the crowds arrived, went back for a mid-day nap, then headed out again until after dark.

Since YS is so big, you'd do best to split your time between 2 locations if you plan to also spend a night or two in the Tetons. Or 3 if you'll be in YS for all 8-9 days.

We've been to YS in every season except spring and loved it each time. It is by far our favorite National Park. So come and enjoy it. You might find yourself adding it back to your bucket list for a return visit.