Before learning how to get into Bhutan, it is important to note firsthand that it’s mandatory to use local registered tour operator to make arrangements for your travel to Bhutan except nationals from India, Bangladesh and Maldive.
There are only 2 ways to get into Bhutan – through Air and Land. Bhutan being the land locked country it topographically rule out the waterways. The most popularly means is through Air as it is well connected with other tourist destinations. There are sizeable guests who likes to venture to travel through land also. It is mainly seen that the cases arise when they wish to integrate their Bhutan trip with Darjeeling, Sikkim and Assam and vice versa.
Air.
All the international  flights lands or take off from only international airport in Paro. To ease the air traffic in Paro, projects in Gelephu ( mid-south border town)  to construct second international airport is underway. Bhutan visa which are send over email are stamped upon arrival in Paro airport.
Druk Air & Bhutan Airline are the 2 carriers operating currently.  Druk Air specifically have wide networks  with flight connection into the regional countries. Its operates its flight daily both ways from Bangkok (Thailand). It also flies twice or thrice  weekly from Dacca (Bangladesh), Kolkata, Bagdogra, Delhi, Bodh Gaya,Guwahati, Mumbai in India, Kathmandu in Nepal, Singapore.( Please check the Druk Air website for the schedule)
Bhutan Airlines started its operation in Oct 2013 with the daily flight both ways from Bangkok and Kolkatta. Recently they have included Delhi also in their operation.

Land
There are three port of entry and exit while travelling overland. The widely used is the Phuentsholing,  (border southwest commercial town), roads and railways networks are well established with other popularly tourist destination namely Darjeeling, Kalimpong & Indian state of Sikkim and rest of India. And it’s merely 4 hours drive to Bagdogra (nearest Indian airport). Jaigoan is the Indian border town where you have to exit Indian visa to enter Bhutan and vice versa.
Also you can enter or exit Bhutan from Samdrup Jongkhar( far south east town). It share its border with Indian state of Assam. The nearest Indian Airport is at Gawahati, which takes approx 3 hours to reach Samdrup Jongkhar. It adjoins with small Indian border town called Daranga, which harbour Indian Immigration Checkpost for related Indian visa work
The third and last port of entry and exit is Gelephu ( Mid south border town.). Which is 245 kms away from nearest Indian airport of Guwahati.  Dathghari is the adjoining Indian border town, where all relevant Indian visa work are processed by the Indian Immigration Checkpost.
The Bhutan e-Visa send over emails are stamped upon arrival on these border Bhutanese town. And it is important to note that all the visitors travelling overland requires Indian visa.
For clarity in term of Indian visa, more specifically recently floated Indian e Tourist visa on arrival. Please find some time to read carefully as it has thrown confusion at large and can potentially pose serious problem.Please click on the subsequent hyperlink   Single entry India E-Tourist Visa Vs Regular Visa