You can travel to Bhutan via air or road. There is a direct flight from Delhi to Paro else you can also reach Bagdogra Airport and then proceed to Phuntsholing, the border town between India and Bhutan. You can also make it through train. The nearest railway station is New Jalpaiguri. From there, you can hire a taxi upto Jaigaon or Phuntsholing. Remember to keep your Voter ID with you as it is a mandatory requirement for getting issued the tourist permit.
Most travellers fly to Paro, bhutan's only international airport through Druk Airways. Most flights. Flights to Paro are from Bagdogra/Guwahari/Kolkatta/New Delhi. Indians have the luxury of reaching Bhutan by road through one of its 3 entry points - Phuntsoling, Geluphu or Samdrup Jongkhar. This is only an entry permit, a further extension may be obtained in thimpu. If you are travelling by your own vehicle (Indian registered) a route permit must be obtained travel authorities in Bhutan as well.
You can go by car,train,flight or bus. The distance from Delhi to Bhutan is 1600 KM shortest route is through Agra, Lucknow, Darbangha and Silliguri. I advise you to go by flight as it saves time it takes just 3 hours to getting to Bhutan's Paro International airport or you can indirect flights via Kathmandu(Nepal).