The easiest roads are from the south via I-80 to 287 or 191 into Grand Teton Nat'l Park and continue north into Yellowstone.
Also coming in from Idaho and West Yellowstone is easy. Same goes from the north via 89.
Coming in from the east is the most difficult via Cody, WY or via Cooke City.
You ARE planning to tour Grand Teton Nat'l Park, aren't you? I'd definitely advise it. They are both completely different in looks and things to do.
Read each national park's website thoroughly for lots of good information on the roads, camping, things to do, road maintenance/closures, etc. There are always updates so check again just before your trip.
http://www.nps.gov/yell/index.htmhttp://www.nps.gov/grte/index.htm