Hiya,
Yes, the best way to reach Phuket from Chiang Mai is to fly; it's likely that Air Asia (a budget airline) could be substantially cheaper than Thai Airways, especially if you book in advance (i.e. asap!). They have a 20% off sale on right now (until 13 January) so prices will be even more appealing. When you do a dummy booking to see how much seats would cost, ensure you've gone all the way through and that all fees have been added, including checked cabin baggage etc, as like any budget airline they charge for every little thing.
Be aware that Aleenta and other such luxury resorts are generally out of the main tourist areas out in the sticks. This is great for that "getaway" feel but it also makes you a "prisoner" when it comes to high transport costs, high meal costs etc. Aleenta "Phuket" in particular is miles out from that kind of thing - it's not even on Phuket island, but rather on the mainland north of the island! To answer a previous question of yours, it's dead on the beach - this is the satellite view of it:
8°15'16.77"N , 98°16'50.35"E - Google Maps
With that in mind you might wish to spend your "other" days down in tourist central at southern Phuket - Kata, Karon, Rawai or Patong etc. Lots more chance to do activities in this area also, I'd imagine. There are quite a few suggestions on this forum for such places, and I am sure others will be able to chip in with suggestions too.
I have booked a few days in Phi Phi in February and will be staying at a nice resort called
Zeavola... a little bit of "splurge" from me too

I will post a review of the place here but this won't be until the end of next month. You can get to Phi Phi easily enough by ferry from southern Phuket.
As for trustworthy (booking) sites, I primarily use
hotels.com - they have a 10% discount code at the moment,
HCJAN10 - or
Agoda. One thing to watch with Agoda is that taxes are not immediately included in the price quoted, which I think is poor show, whereas with hotels.com the price you see is the price you pay (in the UK at least).
HTH
cheerio,
Steve