You can travel by road to any of the four provinces from Bangkok. A drive from Bangkok to Pak Chong is most reccomended. From Pak Chong to the park, there are songthaew, taxi and motorbike services available.
You have to reach Bangkok via air from your respective city, then drive to Pak Chong, and then a songthaew to the park.
You can take a bus from Bangkok to Pak Chong which takes 3-4 hours and costs 150 Baht.
The roads in Khao Yai National Park are extremely well maintained. The best way to look around is to rent a motorcycle from Pak Chong costing around 300 Baht per day. Some travellers often hitchhike in the park too.