Yellowstone NP

If you don't have the cash for a bus ticket, hitching would be the quickest/easiest option.

Else you could take the BNSF to Laurel (Denver > Cheyenne > Casper > Laurel), then take an MRL to Livingston/Bozeman. Of course you'd still need to hitch from there.
 
This is an old ass thread but j wanted to mention that u can hitch to the park from colby instead of riding all the way up to laurel. I hitched from laurel cuz i wanted to ride up there n it was p easy. Beautiful ride long ass sidings tho