Take a ride east, cross the mountain range and visit Park City. It´s a great ski area and nice city. Just half an hour away from Salt Lake City. Some Olympic ski races took place in Park City in 2002.
The mountains are always close by. You can hike to peaks even in the springtime and snowboard on the hillside. This picture here is of a friend of mine boarding on the hillside up Farmington Canyon- about 20 minutes drive north of Salt Lake downtown. Farmington Canyon is a great place to go if you want to hike and ski/board.
SnowBasin Ski Resort is by far the best resort in Utah if you like to Ski or Board. It is usually $30-$45 per ticket... but all worth it! This is a picture from their website of the mountain.