In addition to the snow that might come down, you also have to deal with all the winter's worth of snow on the ground. In Yellowstone, that is quite a bit, usually. I don't know if you are riding out for a summer job or what - but most of the entrances don't even open until May - and are often closed temporarily because of May snow.The only entrance that is open year round is the North Entrance at Gardiner, Montana. The East Entrance is scheduled to open May 7, 2010 - the South Entrance on May 14, 2010 - weather permitting.
If you are going to be able to do it later - arriving in May, you could ride from Scottsbluff, Nebraska to Casper, Wyoming - then to Shoshoni, Thermopolis, Basin and Cody. This would keep you as low as possible in elevation. If you were any earlier, you would need to ride into Montana and then down into Yellowstone from Livingston to Gardiner.
Something tells me you are not in any way prepared for this. Spring blizzards can be deadly. It's 75 degrees one day and 15 degrees then next with a wind chill of negative 20. It can be done, but only with extreme care by someone who is experienced and well-informed. The fact that you are not even aware of the opening dates for Yellowstone roads suggests that you don't have the appropriate background.