Most travellers transit in Loei town, but we were told that at least three buses depart from Bangkok's Morchit (Northern) terminal and terminate in Chiang Khan. Several daily buses also run direct from Nakhon Ratchasima, stopping in Loei. At least one bus returns to Bangkok from Chiang Khan around 6:00 with the rest departing around 18:30; they pick up at different places in town so ask your guesthouse for help, or just head to Loei first.
At Loei bus station, the early bus from Nakhon Ratchasima departs for Chiang Khan around 6:00 in the morning, another around 11:00, and the others running roughly every hour from 14:00 into the early evening. The fare is 38 baht. You can also take one of the large songthaews that run frequently from Loei bus station until 16:00 for 30 baht.
In Chiang Khan, buses to Loei town (and onwards to Nakhon Ratchasima) depart from a small station off the east side of Route 201, within walking distance of Sri Chiang Khan Road. Three buses depart between 6:30 and 9:00 and again between 11:30 and 15:30. Songthaews to Loei pick up on Sri Chiang Khan Road, just east of the intersection with 201.
At time of writing, songthaews to Pak Chom are limited due to ongoing road repairs. Your best off heading down to Loei and taking the local bus that departs every morning at 6:00 and stops in Pak Chom before carrying on to Sangkhom and Nong Khai. Otherwise you could take a tuk tuk.
Chiang Khan is a small town that's perfect for walking or bicycling. Loads of shops and guesthouses on Chai Khong Road rent out bicycles; the going rate is 20 baht per hour or 100 baht for a full day. Many guesthouses provide bicycles to guests for free.
Otherwise, tuk tuk is the only option, and there are plenty of them to go around. We were repeatedly told that no motorbike rental is available in Chiang Khan. One guesthouse owner said this was due to a foreigner crashing at some point, though we suspect the tuk tuks might have something to do with it.