By Flight - There are three flights per week to Udomxai from Vientiane, which costs about 100 USD per person.
By Bus/minivans - There are two bus stations in the town being connected to all the major cities. Buses leave from Luang Namtha, Pak Beng and Nong Khiaw, which are the nearest towns, which takes about 45,000 to 60,000 LAK per person and takes about 4 to 6 hours.
From Luang Prabang, there are buses thrice every day leaving to Udomxai, which costs about 60,000 LAK per person and takes about 5 to 6 hours.
The best way to explore Udomxai is on foot or by renting a motorbike. For getting to caves and nearby villages, you can reach by hiring a Tuk-Tuk or renting a bicycle daily. Cabs are available too if you want to commute to faraway places more comfortably. You can also take up boat rides to the nearby villages.