Bhutan has no railways, no domestic commercial flights to speak of, and only one real highway -- the east-west lateral road that winds across the entire country. Getting around means driving on mountain roads, and you should adjust your expectations accordingly.
By car with a guide -- If you are on a tourist visa, your tour operator provides all transport, typically a comfortable SUV or minibus with a driver and guide. This is by far the most practical way to travel. Distances that look short on a map take hours on Bhutan's winding roads. Thimphu to Punakha is only 77 kilometres but takes around three hours. Thimphu to Bumthang is a solid two-day drive. The roads are well-maintained by Himalayan standards but narrow, with steep drops and occasional landslides in the monsoon season.
By public bus -- Bhutan has a public bus network connecting the main towns, but buses are crowded, slow, and not particularly comfortable. They are used primarily by locals and Indian nationals. The views from the window are extraordinary, but you will share the ride with livestock, sacks of rice, and more passengers than the bus was designed for.
By taxi -- Taxis operate in Thimphu and Paro without meters, using fixed rates for routes within town. Negotiate the fare before getting in. For longer journeys, you can sometimes hire a taxi for the day, arranged through your hotel.
On foot -- Many of Bhutan's best experiences require walking. The Tiger's Nest hike, the Bumthang temple walks, and virtually all trekking routes are on foot or with pack horses. For the adventurous, multi-day treks like the Jomolhari Trek and the Snowman Trek (considered one of the world's toughest) are extraordinary but require serious fitness and advance planning through your operator.
By mountain bike -- Cycling is gaining popularity, especially in the Paro and Thimphu valleys. Some tour operators offer dedicated cycling tours. The mountain passes are brutal climbs but the descents are exhilarating.