jlunsetter,
The directions you've gotten going through Red Lodge to Yellowstone is indeed a great route for scenery, but it's going to be a LONG drive. If you decide it's too far, here's a couple other suggestions:
Instead of leaving Yellowstone and driving north to Livingston, exit the East Gate and drive through Cody, then take Hwy. 16 east from Worland and drive through the Wyoming Bighorns. They're lovely mountains, and you can hook up with I-25 at Buffalo and head to Denver from there.
Another idea, if the mountains and scenery don't have to be through Montana, would be to head south from Belfield straight to Spearfish, SD. From there you could take a brief tour of the Black Hills, Custer State Park and Devils Tower, Wyoming, then west on I-90 until you reach Hwy. 14, take it through Ranchester, WY and over the Bighorns, then head to Worland and pick up Hwy. 16 and head east back over the Bighorns to Buffalo, WY and back onto I-25.
Just a couple more options... from a NE Wyoming guy.

Montana's Rockies are much further west than those in Colorado or Wyoming. But the drive from Red Lodge to Yellowstone is hard to beat no matter where you go.