I grew up in the SLC, so I'm very familiar with the area's ski resorts. As the previous poster suggested, both Brighton and Solitude are great for beginners but also offer terrain for more advanced skiiers. They're also great for snowboarders. Snowbird *does* have some of the best expert terrain in the area, but they've also got some nice intermediate and easier terrain IMO. After learning to stand up on skis at the Brighton, I honed whatever skills I had acquired on the Gad Valley trails at Snowbird. Snowbird also has the best lodging ammenities outside of the Park City area.....the previous poster was correct that the Park City area will be more expensive than the Cottonwood Canyon resorts listed above, but for ski town ambiance it can't be beat.
Were it me, I'd consider staying down in the city--this would give you the most skiing options and there's tons of good places to eat in Salt Lake. You'd probably also save some money on the lodging. Keep in mind that from downtown SLC you're right around 30 minutes from any of the aforementioned resorts.....