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Diana,
If you trust the recommenations of a "skipapa", I'm happy to share my thoughts (I'm a ski-loving, stay-at-home dad, if that increases my credibility in any way). Last year we had our 4-year old daughter at both Park City and Deer Valley kid's ski programs. Both are fantastic, she had a great time, and she's hooked. There are just some small differences between the 2 in terms of price, inclusion of equipment, # of hours of instruction guaranteed, and teacher/student ratios....so check the resort websites to compare those details. You really can't go wrong with either. And both mountains have beginner-friendly areas where you can ski with your kids near the base, plus plenty of green terrain up the mountain once they're ready to ride the lifts. What's fantastic about the area is that you can ski both mountains during your visit, since they're literally 5 minutes from each other, with free parking at each base area, plus the free Park City bus takes you to/from both. We stay at Lakeside Condos, which are within walking distance to Snow Park base lodge at Deer Valley...or you can drive in 60 seconds, or the condo's have a semi-private shuttle to take you to/from your door. My only preference for our beginner skier is that she doesn't have to worry about snowboarders at Deer Valley. But I know she'll be skiing both mountains for years to come. If you do plan to ski with your kids, I highly recommend the ski vests with the integrated handles on the back...they REALLY help with pulling them up (and holding onto them) on the chairlifts...plus some have long straps you can use to give them some space to learn their basic slowing/stopping/turning skills, but still allows you to rein them in if they gather too much speed.
Have a great time.