Where are you flying from? Most flights from CM to anywhere outside of the country go through Bangkok (with a few exceptions), so I think the order doesn't really matter that much in terms of flights. That said, if you are planning on shopping, plan Bangkok to be your last stop so you don't have to haul things around with you.

With four days in Bangkok, I'd stay in the city (I wouldn't go to the Tiger Temple in any case, but that is a different issue). There is more than enough to keep you busy in Bangkok. The one day trip I would consider would be to Ayutthaya, take the boat one way.

For me, four days in CM would be deadly. Some people love CM, but there just isn't enough to keep me interested there. And if you have any interest in hilltribes, you will need an overnight trek to see any who are still living their traditional lifeways. The ones tourists get driven to are more in the range of human zoos. You might consider going farther north into the Golden Triangle area. I also find the food not nearly as good as in Bangkok.

So I might tweak your itinerary a bit, but I do think you have the right idea.

me too days in CM unless you have specific ideas for visits..

i doubt any flts you want leave from CM to home..

Another reason to avoid CM is that February to April is the so called burning season where farmers both in the north of Thailand and in Laos slash and burn forest and vegetation to create farming land. PM-10 levels which are a measure of how clean the air is swiftly jump to danger levels during that period. You can check current levels at http://ourchiangmai.com/air-pollution-in-chiang-mai-thailand/

If you want another island to visit, take a flight from Phuket to Koh Samui which is only 55 mins away. Flights are quite expensive because the airport is privately owned by Bangkok Air and that's reflected in the air fare. Check the Bangkok Air site for ticket prices: http://www.bangkokair.com/eng

As regards your trip from BKK to Phuket, do you want to fly with a budget airline, or a regular flight? If it's the former, you'll need to go to the budget airport which is Don Mueang. Budget carriers are either AirAsia: http://www.airasia.com/ot/en/home.page or Nok Air: http://www.nokair.com/nokconnext/aspx/index.aspx

Otherwise you can fly from Suvarnabhumi which is the international airport with either ThaiSmile, or Bangkok Air. But fares are about double those of the budget carriers. Here's the ThaiSmile site: http://www.thaismileair.com/

We rearranged a trip to avoid the burning of the hills after reading about how bad it can be. We were in NThailand this past January and they had already started the burning and some areas were thick with smoke....not CM yet, but you could see smoke in the distance just outside of CM. Aside from it being horrible health wise, it will limit your views of the area.

In terms of time spent in each place, it really depends on what you are interested in. Obviously you have the beach and if you are beach lovers you might want to explore the beaches for a day or two longer. BKK is a huge city, we thought it was like NYC on steroids. Lots to c and do and fabulous food. It has a great transportation infrastructure and easy to get around despite its size. Chiang Mai is a smaller town and we thought way more charming. It has a much more small town feel and is very laid back. We loved walking all over and discovering the various neighborhoods. You could also go up to Chiang Rai. Our long TR describes these places
http://www.fodors.com/community/asia/our-seasia-odyssey.cfm.