We have crossed Nebraska a number of times on trips west. We usually enter at Nebraska City and take route 2 to join I80 at Lincoln. West of Grand Island we move over to route 30 to get away from the heavy truck traffic on I80. Route 30 is a bit slower, but a much more relaxed drive. At Ogalalla we pick up route 26 all the way to I25 in Wyoming. Route 26 follows the Oregon Trail past Chimney Rock, Scotts Bluff, Fort Laramie and the wagon ruts at Guernsey. Fort Laramie and the wagon ruts are well worth a stop. At Casper we pick up route 20/26 to Shoshoni then route 20 trough the Wind River canyon to Thermopolis. From Thermopolis route 120 will take you to Cody for the east entrance of Yellowstone. This route has no mountain driving until the east entrance.
Regarding your campground question, Fishing Bridge is the only campground inside the park with hookups. If you need hookups, that's your best bet. If you can get along dry camping then my suggestion would be Madison campground on the west side of the lower loop. The great majority of the thermal features are between Madison and Old Faithful, plus it isn't a long drive to the Yellowstone Canyon. IMO Grant Village is too far from the action.
