The easiest way to go around and explore the area or get inside Khao Yai park is by renting a motorbike or a car if you do not have your own vehicle.

Car: Best rates for car rentals are available in Bangkok. All international companies are present in Thailand, remember that choosing a company with proper insurance is important if you are a tourist and that your own travel insurance probably does not cover you in case of trouble while driving any type of vehicle in Thailand. here a few links:
Budget Car, Thai car hire, Thai rent a car

Motorbike: As songtaeow drives along only Thanarad road, renting a motorbike is a good option to discover the area, specially some nice country side, area as The Asoke Valley where you will find the vineyards of the area or enjoy local dining and shopping.
Drive safely of course, while Thanarad can be a busy road, most side roads are quiets and a good way to enjoy peaceful country life.
Motorbike are rented along Thanarad for around 500 Thb a day, a good option is to shop it in Pak Chong, KhaoYai Motorcycle runs a good shop in Pak Chong and have an excellent fare at 300Thb a day for reliable motorbikes . Khaoyai motorcycle website