Sylvia,
I would book your flights first. Have you tried airasia.com? They are my preferred budget carrier in Thailand. The site you referred to is probably just an agent for airasia.com.
I am not sure when you are going but I plugged in dates two months from now and got about 2000 Baht one way from Chiang Mai to Phuket including taxes (but not including baggage). This is about $66 - seems like a bargain to me! They run off the budget airline model so as more seats are booked on the plane the seat price quoted increases. So book your two flights now (separately), then snap up your accommodation too.
Enjoy the planning - it's half the fun!