Well, in Yellowstone, all the roads will still be open (weather permitting) except the road between Canyon and Tower Junction. This would mean you could visit all the natural features (Grand Canyon of the Yellowstone, the geyser basins, the hot spring terraces at Mammoth) except Tower Fall and see wildlife in Hayden and or Lamar Valley and along the roadside all over the park.
All lodging will be closed and all food facilities except the Geyser Grill fast food at Old Faithful, the Canyon Adventure Store which may have very limited food (maybe hot dogs? sorry don't really remember!) and the Mammoth General Store which has some fast food type stuff.
For your 2 nights lodging you could stay both nights in West Yellowstone or you could do your first night in Gardiner and the next one in West Yellowstone.
Here is a suggestion for a possible itinerary based on staying the first night in Gardiner and the 2nd night at West Yellowstone.
Day 1: Enter thru the South entrance. Stop at West Thumb. Head north around the Lake, stopping at Fishing Bridge. Park and walk across looking for wildlife. Stop at Mud Volcano, then the South Rim going to Artist Point for a view of the Lower Falls. Return to the road, do the North rim, stopping at one overlook, perhaps Brink of the Upper Falls. Stop at Canyon Visitor Center for a pit stop if the bathrooms are open (VC itself is closed). Go to Norris Junction, then head North to Mammoth. The better bathrooms here are the ones next to the gas station and bus parking lot. Depending on how your time is running, walk the lower terraces or just drive out the N. entrance to Gardiner.
Day 2: See the Lower Terraces if you ran out of time yesterday. Head south. Go to the Upper Geyser Basin first which will be the furthest away, then work your way back to Madison Junction seeing the other geyser basins. When you get to Old Faithful, go to the Visitor Center (it is still open) and check on the eruption times of predictable geysers. See if any of those would coincide with a walk thru the geyser basin. As you head to West Yellowstone, work your way back thru the other geyser basins.
Day 3: Leave West Yellowstone and head to Moab. According to Google Maps this is about 9.5 hours/552 miles.
I recommend you get a cooler with food and beverages for your lunches.
Make sure you have paper maps to refer to in case you run into bad weather and have to make a change of plan. You cannot rely on a GPS for this. You also need to be checking weather every evening for the next day and be ready for a Plan B if a storm is going to move thru.
We would love to have a trip report.