Come see us in Breck!As viking kaj mentioned, Breck is a good choice if you're thinking CO. Breckenridge has lots of good learning terrain, and quite a few EpicSki Bears who teach at our Ski and Ride School (including yours truly), including (at least) one of our ski school supervisors.
Also Breckenridge is (pardon me while I slip into advertising-speak) "The Perfect Mountain Town" (TM). After you finish barfing from the ad-speak, the reality is the town is great, with lots of shops, cafes, a theatre company, music, lots of fun stuff. Since we're owned by the same company as Keystone (Vail Resorts), you also can ski Keystone on your lift ticket. If you buy a multi-day pass, which you probably would coming out for a ski week, you can even do a day at Vail (about 35 miles away).
If you decide to hit Utah instead, that's a good choice too. You can go sample a lot of areas. I'll second the recommendation about Brighton - it's a old-style feeling "ski hill" with a great ski school and lots of friendly staff.
If you do want the "ski town" feel along with the ski vacation, not just the on-hill experience, then probably staying in SLC won't give it to you. Park City would be more like it if going to UT. As would Breckenridge (with slightly less glitz than Park City). Or really anywhere in Summit County - a suggested up-thread, staying in Frisco, which has an old Main Street feel while convenient to Copper, Breck, and Keystone by quick drive or free bus, can save some money. (Well maybe not staying "anywhere" in Summit County - Silverthrone motels are probably cheap but I like to think of it as our little piece of New Jersey in the mountains - gotta put the strip malls somewhere.)
Our colleagues/competitors over at Copper have some great instructors too, lots of them who post here at epic and some who already jumped in on the thread. One of the nice things about Summit County is that you can sample several areas. Also true about the SLC area.
One other place you might like, though it depends on how you'd be getting there and how expensive the trip might be, is Whitefish, Montana, Big Mountain. Whitefish is a great western town and Big Mountain is not at all crowded. Lot of varied terrain, including a lot of good intermediate stuff. If you want to make an adventure out of it, the trip there on Amtrak on the "High Line" route which takes you right into the center of Whitefish is an interesting way to get there. (Interesting = "great rail journey" or "absolute horror and waste of time" depending on your opinion of rail travel).