I don't think there is any bus service to the local areas for Ogden. Powder Mountain and Snowbasin. Which are each about 20 minutes away depending on where you are in Ogden. The roads are pretty good up Ogden Canyon, which is the way I would recommend getting there. Ogden Pass or whatever it's called is very slow going in comparison and you don't gain anything except different scenery by driving over it.
Both Powder Mountain and Snowbasin are well worth your time to visit and generally less crowded then the resorts to south by Salt Lake City.