I was the opposite of the above. My dad bought me my first horse when I was your daughter's age. I'd had very little experience beforehand, though I had ridden family horses from time to time. (I DO suggest lessons even if you buy her the horse, if only to begin building her confidence)
My dad knew a lady down the road who we'd met a few times before, and he asked if she would help us find a horse suitable for me. She went with us to all the horses we looked at and rode them first, and finally we found one we all thought would be good for us.
Unfortunately, he did have shipping cough, but after his quarantine period and treatment by the vet, he became my best friend. I rode him everywhere.
My suggestion would be to first be sure of your finances. Set a budget (a realistic one) for a first time horse. Don't scimp on the money for a good beginner horse. They are priceless. It would be in your best interest to ask an experienced rider for their help. And GET A VET CHECK BEFORE BUYING!!
You don't want to get sucked into a longterm loss of money as far as health goes. Also, be sure the seller has ALL the right paperwork for the horse you decide on, and even requesting a trial period where you come and ride the horse for a week wouldn't be a bad idea.
As far as a good breed, I like Quarter Horses for their all around disposition and ability. Most people would suggest a gelding, as they tend to be most laid back, but I have never had problems with a well picked mare. I would not suggest buying anything under the age of 6 and anything as old as 18. As far as size goes, that really depends on what your daughter is comfortable with. I prefer a horse in the 14.2-15 hand range.
Also, be aware that taking your daughter horse shopping will be DANGEROUS. haha. So take caution that the pretty ponies will be alluring, but don't buy on looks alone! Look for the right temperament, a seasoned horse that has experience with kids (lesson horses are awesome for this), and a horse that is willing and curious, yet not easily shaken.
Being picky will save you both a lot of headaches and a lot of heart break in the long run.
Feel free to ask any questions! 14 years in, and I still ask questions. You can never be too prepared or know too much when it comes to horses.
Riding: The art of keeping a horse between you and the ground.