The only airline operating into Bhutan is Drukair (Bhutan National Airline) and Bhutan Airline (Tashi Airline). It operates from Bangkok to Paro via Kolkata, Bagdogra, Gaya, Guwahati, Dhaka and Delhi to Paro via Kathmandu. Apart from Paro, there are three more domestic airports in Bhutan: Bumthang in central Bhutan, Yonphula in the east and Gelephu in the south. As of now, Bumthang, Yonphula and Gelephu flights are operated by Drukair 3 times a week. (Tuesday, Thursday and Sunday).
Entry / Exit to Bhutan are also possible through Phuentsholing, the southern Bhutan frontier town and the main commercial hub. Bagdogra airport, in the state of West Bengal (India) is the nearest airport about 5 hours (175km) drive from Phuentsholing. Also Bagdogra (and its main town Siliguri) is the convenient point for visiting hilly region of Darjeeling, Kalimpong and state of Sikkim.
Badrapur (about 6 hours drive from Phuentsholing) and Birat Nagar (about 7 hours drive from Phuentsholing) are two other airports in Nepal, connected with regular flights from Kathmandu.
You may also choose to enter and exit Bhutan through south- eastern district of Samdrup Jongkher, the border town with Indian state of Assam. This requires you to drive all the way to Eastern Bhutan or take a domestic flight to Yonphula and exit into India or do in reverse order and exit from Paro airport. The approximate drive to Guwahati (capital town of Indian state of Assam) is 3 hours (180km) from the boarder.