I take my kids to Al Ahli Hospital. Dr. Attiya (this is her first name, can't think of the last right now) is great. She speaks perfect English and is just a great doctor. Also there are Dr. Manosh and Dr. Mamood. Both very good as well.
974-489-8888. Open until 9pm.
For really late night and emergencies, the public children's emergency place is very good too. A little different than what you're used to but they are very thorough.

A great dentist for kids and adults is Dr. Alex Gibson at the Dr. Sarah's Dental Centre. He's Scottish and has a great manner with the kids. 974-444-1050/70. Located off Al Waab near Sadd Exchange.

If you haven't already, check out the Qatar chat board at expatwoman.com for lots more mommy info.

Good luck with the move.