Once you're at West Yellowstone and want to get to Grand Teton NP the best way is through Yellowstone and use the south exit into the Tetons. Try to leave early morning to avoid the crowds and to see more animals along your drive.
Look at Yellowstone's web site (& any other national park you'll be visiting) for much good information on roads, road closings, special alerts, camping, things to do.. and more.
https://www.nps.gov/yell/planyourvisit/campgrounds.htmand driving through all entrances:
http://www.rvtechmag.com/travel/4_yellowstonebyrv.php