Walkthrough: With all the barriers removed, the path to the race track in Shiveria Town will now be open. By speaking to the Shiverians once more, you will learn that because Bowser stole the Frost-Frosted Cake, the new prize for winning is a Multi Moon. So head down the path to the starting area for the race. As you arrive, you will learn one racer won't come out of his nearby room, and that he is needed for the race to begin. Because of this, head into the next room and go up the platforms - getting the Purple Coins by clearing out the snowdrifts - reaching the center of the room and capturing the Shiverian with the help of Cappy.
As the captured Shiverian, go up into the next room to find some Purple Coins. From there, head to the race track. As you try out training for the race, you should find that you'll gain more momentum by bouncing at the moment of the impact, the angle of your landing determining where you'll bounce off next. Get the hang of things on the fly, and then use your training to win the race and obtain the Multi Moon.
Once you have, your time in Snow Kingdom will conclude. Returning to the ship will reveal the blizzard has now stopped, as well as a few new things having been added to the kingdom. If you still need a few more Power Moons in order to leave Snow Kingdom, explore a little to achieve this. If you've obtain the Power Moons mentioned along the way, you shouldn't need many more in order to leave. Once you have enough Power Moons to leave, you will head off to Seaside Kingdom if you didn't choose to go there first. If did and now have both Snow Kingdom and Seaside Kingdom completed, you will head off to Luncheon Kingdom.