Some people enter Bhutan by road at Phuentsholing, which is also the border town with West Bengal. If you’re in Bhutan for a short time and want to see Sikkim and Darjeeling as well, this might be a good option for you. From Thimphu to Phuentsholing takes about 5 hours and another 6 hours from Phuentsholing will take you to Sikkim/Darjeeling, which is called the Queen of Hills.
You can enter and exit Bhutan at Samdrup Jongkhar, a town on the border with Assam. This will save you a long drive back over the mountains to Paro after visiting eastern Bhutan. From Samdrup Jongkhar, drive 110 kilometers (about 68 miles) to Guwahati, from where you can fly to Kolkata, Delhi, Bangkok, or Bagdogra or get a train connection to various Indian destinations.