I would look at the Shilo Inn downtown or the Best Western Garden Inn in downtown Salt Lake City. The SLC airport to downtown is about 10 min. by taxi and one or both of the hotels mentioned may have limo service. You would not save more than five minutes of drive time by staying closer to the airport. Don't worry about being too far away from Park City by staying in downtown. Either the Shilo Inn or the Best Western Garden Inn are very convenient to freeway entrances which will help you get on the slopes early. There are other hotels east of downtown, but their freeway access is more difficult. The road to Park City is interstate highway ( I-80 eastbound). It is nearly always cleared before any surface roads so even if it snows, you shouldn't have trouble getting to Park City. The same can not be said for either of the Cottonwood Canyons where Snowbird, Alta and Brighton ski resorts are located. Those highways can be closed for hours if there is a big storm. Your total drive time from downtown to the Park City Mountain Resort would be about 45 min. (assuming dry roads).
You said you wanted to find a "nice place in town" the next day. When you say "in town" are you looking for Park City or Salt Lake City?. If you are thinking Park City, you will want to make reservations soon. Depending on which weekend in February you are coming, you might have a little bit of difficulty getting rooms if you wait too late. President's Day weekend is very busy. And Park City schools have a "winter break" in February so all the locals encourage out of town visitors to come during that break. I'd make reservations soon.