Direct international flights to Chiang Mai from:

Hong Kong:  Dragon Air /Cathay Pacific, Air Asia and HK Express
Kuala Lumpur and Macau: Air Asia
Luang Prabang/Vientiane:  Lao Airlines
Seoul:  Korean Air
Singapore:  Silk Air and check plans for Scoot ( former Tiger Airways)
Yangon and Mandalay:  Bangkok Air
More direct flights from China keep opening up:
Beijing, Shanghai and Kunming: Air China, China Eastern, China Southern

Domestic flights:

Thai Airways and Thai Smile has several daily flights from Bangkok’s Suvarnabhumi airport:
http://www.thaiairways.com/en_TH/index.page
https://www.thaismileair.com/en

Bangkok Airways often good web fares from Suvarnabhumi airport and enjoy their boutique lounges:   www.bangkokair.com

Domestic Budget carriers:
from Bangkok Old Don Muang airport:
Air Asia:  www.airasia.com
Nok Air:  www.nokair.com

Chiang Mai Airport is just a short 10-minute drive into town.  Reasonable fixed-price airport taxis are available.

Train Travel Bangkok to Chiang Mai:
approx 12 hours via Ayuthaya and Phitsanulok where you can jump off for Sukhothai about one hour drive away.
And coming from Hualamphong Station in Bangkok, the train stops right at the Old Don Muang Airport.
Train buffs may enjoy the ride, new train carriages have been introduced but there are still often delays.
With so many cheaper flights now, it is so easy to fly and spend the time exploring scenery around Chiang Mai.  Who knows what the landscape may be like if the planned High-Speed Trains are finally introduced in years ahead.

More Good Train Info within Thailand on:
https://www.seat61.com/Thailand.htm

Bus Services into Chiang Mai:
There are 2 Bus Stations in Chiang Mai.
Arcade Bus Station is divided into 2 terminals:
Long distance buses of all standards from simple air-con to Luxury VIP buses.
Bus and Mini Vans to Mae Hong Son.
Travel time by bus Bangkok to Chiang Mai is approx 9 hours.
From Bangkok’s Mo Chit Bus Terminal you can travel anywhere in Thailand cheaply.

Chang Puak Bus Station
just north of the old moat is where you get local buses and mini-vans to:
Tha Ton via Mae Rim, Mae Taeng and Chiang Dao.

When to Visit:
Most pleasant times are from  mid October – mid February when days are usually cooler.
October is the end of the rainy season with skies clearing and lush green landscapes before the peak season crowds starting to come in November.
December and January can become quite cool in the mountains.  This is also the peak season for travel.

By MARCH-  the burning of the dry-season comes with the smog setting into the whole region of northern Thailand, northern Laos and Myanmar-  so this is the time to AVOID.
April and May are the HOTTEST months.  Songkran and Thai New Year water throwing party (the days around 15th April) is a time to join in to cool off if you are into that sort of fun.

June and July more humid but with refreshing rains brings back the clearer skies and there are fewer crowds.
August and September are traditionally the months of heaviest rainfall.
November Full Moon is the Loi Krathong festival when the skies fill with fire lanterns. This has become a very popular and also very crowded event within the city.
The first weekend of February is the beautiful Chiang Mai Flower Festival with flower floats in procession through the city.