If you can get to a UTA stop, you can hop on and be up the mountain in 20 minutes. If you can get to the canyon, put your thumb out, somebody will pick you up, I would (look for the soft top Jeep with the box on top). Pending where you stay, expecially downtown, you might want to rent a car, opt for 4x4 and you won't have to deal with chains or the bus/hitching to get up the canyon on a powder day. Its about a 35-50 minute drive to almost every resort from there (except Snowbasin and Powder Moountain). As far as the road being closed, thats pretty rare, and they work hard to reopen in. Thats usually an avalanche that does that. Can be avoided if you stay in a resort.
Tip: If you ski Snowbird and opt for beers after skiing on the deck, go downstairs and grab a 6-pack from the store (by the elevator), it will save you about 75% over buying them on the deck.