Chiang Mai has an international airport so is well connected with other major international cities. Buses to and from Bangkok take 9 - 12 hours and run at a decent frequency as well.
Direct flights to Chiang Mai are available from all major Indian cities like Delhi, Mumbai, Chennai, Kochi, Kolkata etc. Most popular airlines being Thai and Silkair.
Indians are eligible for visa on arrival at Chiang Mai if their trip is not going to exceed 15 days. All information pertaining to the visa processes are available at the official embassy website.
The Chiang Mai International Airport operates both international and domestic flights. It is located 3km away from the city and is well connected by public transport.
One of the best ways to experience the city and nearby areas is to rent a car and self-drive up the country. Cars may be rented at several places in and around the city with the most convenient being the Airport.
Trains hardly ever arrive on time in Thailand. Local trains run with a slightly reliable frequency. Do check with your hotel about the frequency of trains to and from the city you wish to visit.
One can take a tuk-tuk or samlor to cover short distances and get a feel of the local wind in their hair. To those who consider this a battle between life and death due to the speedy drivers, a taxi would be a safer albeit more expensive option. Local buses best for travellers are B1 and B2 which operate within the city. Songthaew which means 'two-rows' in Thai are modernized mini-trucks which cater to short distances for people going the same way and can be hailed in the street. However, the old city is best explored on foot as it is only 1.5 km sq. in area.