Via Flight - go to Paro
Via train - reach Hasimara from there take a cab to Phuentsholing - the entry to Bhutan. You can also book a cab from here till Thimphu
Hasimara has limited train options so one can reach NJP then hire a cab. Or simply take another train from NJP to Hasimara
Best way is to reach Hasimara and from there board a cab. Indians require a permit to enter Bhutan. Can be obtained at the jaigaon border office or on entry. It’s free and allows one to stay for upto a week.
You can take a direct flight from Kolkata to Paro.
You can also take a train to Hasimara, and then a cab to Jaigaon(India)/Phuentsholing (Bhutan). You'll get the Entry Permit for 7 days from there. If you wish to extend your stay, you can do that in Thimphu. From Phuentsholing to Thimphu, it'll take around 5 hours by car.