When the Oregon Trail
was actively in use as a transcontinental route, it was not much more than some wagon ruts across the prairie. Starting out from Kansas City, in Nebraska it pretty much followed the Platte River which provided both flat ground and water for the livestock. Today that route is the same as used by I-80 west from Lincoln NB. Between Grand Island and Ogallala, US-30 runs parallel to I-80 and will let you get down to 'ground level' so as to see some of the landmarks and towns.
After Ogallala, US-26 follows the route of the old trail northwest from Ogallala NB through Scottsbluff NB to Casper WY. US-25 continues to Moran WY for access to Grand Tetons
National Parks. April could be the best of times or the worst of times. Tourists will not have shown up yet but some of the roads might be closed due to weather. My own experience, in a late October, was what I would wish for you. I had Old Faithful to myself, literally!