There are direct flights from Bangkok to Chaing Mai on Thai Air...

No alrline seems to fly the Chiang Mai-Bagan route that I can see.
There are flights between Yangon (Rangoon) and Chiang Mai via Air
Mandalay (http://www.airmandalay.com/) on Sundays, and also a flight
from Chiang Mai to Mandalay on Thursdays, but Air Mandalay doesn't
seem to offer any flight from Mandalay to Chiang Mai directly. If you
go to Mandalay, you could drive or fly to Bagan (1/2 hour flight or
maybe a 6-hour drive?) or take a ferry down the Irrawadi River to
Bagan, which would be scenic and relaxing but may take a day. The
flight between Yangon and Bagan is about an hour. The quickest way
from Chiang Mai to Bagan would be through Yangon (or shorten the
yangon flight by leaving out of Bangkok via Thai Airways), which would
involve 4 flights and would end up requiring about 4 days all
together. If you opted for that itinerary, you might as well add an
extra day in Yangon to see the Shwedagon Pagoda, Scott Market (Bogyoke
Zei) and perhaps the Reclining Buddha or Sule Pagoda, with dinner at
the Karaweik Restaurant on Royal Lake (Kandaw Gyi)to take in the
cultural show along with the meal. More information on tour groups,
hotels and airlines in Burma can be found at www.myanmars.net.

USC