Alta: Alf's High Rustler is a must. You will know it when you pull up in the parking lot. Stone Crusher, Wildcat Face, Glory Hole, Sidewinder and Supreme Challenge are all very good "named" trails you will find on the map. Ask a local or sign up for the Diamnond Challenge or Mountain Workshop to get a tour and find the hidden stuff. I like making the short hikes into Catherine's Area. Nice trees and plenty of short and steep chutes. Some of those spots in there are pretty secluded so it's best to ski with someone. Your mother may be able to handle So Long. It's not very steep but will be bumped up, powder, or cruded up but I think it's a great run to introduce strong intermediates to off-piste skiing and what earning your turns is all about. Short hikes into Devil's Castle and the Greeley areas are also worth it. Some will say the ultimate hike to and must ski is Mount Baldy's Main Chute. I've never done it. I doubt I could make the hike and then have anything left to ski the chute.
There are plenty of great cruisers located off of every lift at Alta. I'm sure you and your mom will have a great time exploring what I think is the best mountain in the country.
Snowbird: I think Bob Peters did a great report on the Bird and posted it here a couple of months ago. He, Endlesseason, and many others here will be able to provide you with any other info your looking for at the Bird. Have fun.