You will enjoy both of those drives.
287/87 takes you along the wide open valley of the Madison River with mountains on both sides. The route is relatively flat except for a "pass"/hill near Norris. It is a wide road with plenty of space for the slow climbers and speedy passers. You will pass through Ennis which is an interesting town full of Madison River flyfishers when you will be there. You will come to a split where 287 continues east and 87 south. Both will get you to West Yellowstone about equally fast and easy but I would take 287. An interesting stop just before you reach Hebgen Lake is the Earthquake Lake Visitor Center. It shows the story of an earthquake in 1959 which cost a number of lives and dammed up the Madison to form another lake. You will then pass by Hebgen Lake just as you join up with 191 for a few miles into West Yellowstone.
191 takes you north from West Yellowstone towards the Bozeman area through the Gallatin River canyon. It will be fairly downhill for the most part. For a short stretch just north of the 287/191 interchange you are actually in Yellowstone. After you reach Big Sky you will be driving through a narrow canyon along the Gallatin on a winding road that is one of the most scenic in the country. The road is good and you will probably be in a line of traffic that has no choice but to just follow along. It usually moves along at a nice pace though and you will want to just enjoy the scenery anyway. As you approach the northern neck of the canyon there is a construction area this summer that may cause a 15 minute delay while you wait for a pilot car to quide you through the short construction zone. At the north end of the canyon the valley flattens out as you drive past Bozeman on the east toward Belgrade to the north for about 10 miles before you intersect with I90.
Enjoy the area, we love it.