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However you have four cities in the region with airports – Mysore (Mysooru, MYQ), Mangalore (Mangalooru , IXE), Calicut ( Kozhikode, CCJ ) and Bangalore (Bengaluru ,BLR).

The closest possible airport is the one at Mysore , that is approximately 120 km (75 miles) east , in a bit southwardly direction. However the well connected ( and the farthest ) is Bangalore with direct connectivity to almost all major airports in India. Of course both Mangalore and Calicut too are reasonably well connected with rest of the major airports in India.
Except Mysore, all the three are international airports.
Bangalore is some 250km (155 miles) east of Madikeri in Coorg. Mangalore and Calicut are cities on the Arabian Sea coast at a distance of about 135 km (84 miles) and 180 km (112 miles) respectively.

For Mangalore,Bangalore and Calicut , the airports are some 1 hour plus travel from the respective city centers. For Mysore it is about 30 minutes.
If you have booked your accommodation with one of those upscale operators, they usually pick you up from the airports. Check out the details of the package. If you have to arrange the onward travel to Coorg yourself, hire a car. The economical option is to explore the bus option. All the four cities mentioned are well connected with Coorg by road. The state owned KSRTC operate frequent bus services to various towns of Coorg (mostly to Madikeri ) from all the four cities. ( See also Bangalore to Coorg and Mangalore to Coorg for road travel details )
Calicut airport has direct flights to Delhi, Coimbatore, Mumbai, Bangalore, Chennai and a host of international flights to cities in the Middle East.
Mangalore has direct flights to Delhi, Kozhikode, Bangalore, Mumbai, Chennai, Kochi, Thiruvananthapuram etc and also to a number of cities in the Middle East.
Bangalore has connectivity to almost all the medium and large airports in India. Also there are a good number of international flights operate from Bangalore. Thanks to this, Bangalore is the popular choice of airport for Coorg.
Mysore airport has only limited connectivity, mostly as chartered flights.

Many tourists add Mysore too in their tour itinerary. Rather Mysore is a much larger tourist destination and a tourism hub in this part of the country.
Coorg doesn't have direct railway connectivity either. However there are many railway routes encircles the region. This too is an efficient option to reach Coorg, if you are traveling from far away places to Coorg.