Public transport along the King's Highway is neither frequent nor complete, with the stretch between Dhiban and Ariha devoid of any regular public transport. Hitching can be a good way to get around, but be prepared for long waits on deserted stretches. To explore all of the sites covered in this chapter by such methods, you would need up to a week.
One option to consider is the transport (minimum three passengers) organised by the Mariam Hotel in Madaba or the Palace Hotel and other budget places in Amman which runs to Petra in a day, with stops in Wadi Mujib, Karak and, sometimes, Dana and Shobak. Chartering a taxi is also possible for the same journey.
The only alternative is to take a regular minibus from Madaba to Dhiban, charter a taxi to Ariha, then take a minibus to Karak. From Karak minibuses run to Tafila, where you can take another to Qadsiyya (for Dana). Public transport south of Qadsiyya (near Dana) is infrequent, so you may need to take a minibus to Ma'an and then another to Wadi Musa (Petra).