Clustermagnet. If you are on a budget forget about the snowcat. Why bother. There is tons of steep skiing at Alta and Snowbird. It is not hard to find whatsoever. Look at the slopes while you are riding up the lifts. At Snowbird you will see Experts Only Gates all throughout the mountains, most of them are quite doable by an experienced skier. The best way to feel the pull of gravity is to explore and follow the tracks as they go off the trail, into the unknown. I would be careful going where you don't see any tracks.
Regarding avalanches. It seems there has been an increase in in-bounds avalanches the past few years. I wouldn't worry too much, but ski with a friend, they are still a rare occurrence and I would think you are much more likely to die by hitting a tree or getting run over by someone. After storms the patrols will shoot artillery and drop charges on the slopes at risk. The odds are less than 50% that there will even be powder on your trip and it would be dump luck to be caught in one. If you are going to ski the steep and deep you are going to have to take that small chance, that is all there is to it. Don't let anyone blow this risk out of proportion. People die at these resorts every year in all kinds of accidents besides avalanches.
You don't really need specific directions at Alta/Bird, it is all there for you to find, but the advice for skiing the Cirque is good. Look at it while you go up the tram, and probably you would want to traverse down a bit where you will feel more comfortable with the pitch. Ski every lift, there is good skiing just about everywhere. At Alta you want to ski on the Wildcat lift and Supreme. That should do it for you!
Edited by SnowbirdDevotee - 2/1/10 at 2:44pm