Kurt, the single shot is a traditional first rifle...and .22 is the caliber. I still shoot and enjoy my first rifle, given to me by Dad, when I was very young. It's a remington "targetmaster", 1930's or 40's vintage. No scope, iron sights.
It's a shame remington no longer makes them.
Today, single shot .22's come in a wide price range...and styles.
Whatever you select, I would advise against a semi auto as a first rifle.
Here's a place you can begin your research:
http://ocshooters.com/Gen/kidshooting/youth-firearms.htm
For the ultimate .22 rifle? One you could proudly pass down to your daughter when she grows to fit it?
http://www.kimberamerica.com/rifles/kimber1722/customclassic.php