Most people begin the trip to Soppong from Chiang Mai. One option is to fly to Mae Hong Son and continue overland from there: the flight takes about 25 minutes and costs around US$20, and once you land you can rent a 4WD at the airport, take the public bus (roughly two hours, about 35 baht), or arrange a private pickup (around 1,200 baht) or minivan (around 1,500 baht) on to Soppong.
Driving yourself is the most flexible option, letting you stop at sights along the way and travel at your own pace. The road is good throughout, and the trip takes about three hours direct from Chiang Mai — longer with stops, which are worth making. Follow signs for "Pangmapha" rather than "Soppong" along the way, since that's the name used on most road signs and kilometre posts.
The bus is the cheapest option but the slowest: a ticket costs around 100 baht, but most services are non-air-conditioned and the trip takes around six hours, with stops in Mae Malai, Mae Sae, and Pai along the way. A handful of air-conditioned buses also run the route, at a similar price, though without a fixed schedule.