You have the option of taking the I 84 to Pocatello and then heading north through Rexburg, ID to West Yellowstone (Dirct route)...or you could hitch to Hermiston, OR and take the 395 north up toward Spokane and the hitch west on I 90 to Livingston,MT and then south to the park.
I would reccommend the first route. Never done it though. Read through the forums on here and try to find some info. good luck.