TommyGun said:
It's a pretty drive exiting along the E gate (HWY 14) towards Cody. If you're up in the Lamar Valley area you can also exit via the NE gate and go through Cooke City, MT and down the Chief Joseph HWY back to Cody. That is also a really impressive drive and it really shows off a part of the park that not too many people see.
I second this. Follow TommyGun's advice, he knows what's up. Chief Joseph Hwy is awesome and very scenic.
In terms of from Boise to West Yellowstone, I would skip the Interstate as I84 and I86 are not very scenic (read that as not scenic as all) and you will likely either die directly of boredom, or fall asleep and run off the road, roll 14 times and die.
Take I84 east of Boise, exit at Mountain Home (exit 95) and take Hwy 20 east. It will put you in Idaho Falls just 30 minutes or so longer than taking the interstate to Idaho Falls. In all honesty, Hwy 20 is not much more scenic except that it goes directly past Craters of the Moon National Monument which is pretty cool. If you have kids it is a good place for them to get out, run around and maybe climb a cinder cone. There is an extensive lava flow landscape that is pretty cool.
I also like that as you head east from there you go through the barren wasteland that crosses the Idaho National Energy Lab (Nuke Stuff). As you approach Idaho Falls and cross I15 it is like the movie Wizard of Oz...you go from monochrome to color. In about 10 or 15 miles you transition from desert to barley fields, rolling hills and mountains looming on the horizon as you approach Yellowstone.