In general, diving from Phuket is very good, but so is Ao Nang, Khao Lak, Koh Lanta, Koh Lipe and Phi Phi Islands. Some suggest one place, other would say another and the truth might be in between and everywhere. It really depends on who you ask.
I have had the pleasure to do some great diving at all destinations mentioned, but a few dive sites really do sticks out.
From Khao Lak: Richelieu Rock, Koh Bon and Elephant Head Rock (No diving from May 15th - October 15)
From Phuket: King Cruiser Wreck, Bida Nok, Shark Point and South Tip at Racha Noi.
From Krabi: Same as Phuket (except South Tip) and then add Koh See near Ao Nang.
Not everybody would agree, but this is my list. In general these dive sites offers you beautiful coral and great marine life, so bring a camera. Don't forget that some of these dive sites, are not for divers with entry level certification only. If you don't hold a certificate beyond entry level, you can get this in Phuket or another destination with 2 days of training.
When it comes to the best dive shop I'm a bit biased, as I have a dive center in Phuket and Khao Lak, but search around this forum and you'll find some advices on this as well.
Have a great holiday