I agree with Trekker.
Don't expect to see anything. But if you go to any of the parks mentioned by Trekker you will surely find tracks of large wildlife. And that's exciting in itself. Elephant tracks, bear claw marks, wild boar tracks are easily found, and every now and then there are foot prints of carnivores or e.g. porcupines. It shows you're on the right spot, even though it's not so likely that you get to see any of these animals.
I do a LOT of hiking in Khao Yai, but can still pretty much remember every single elephant encounter on the trails in the forest. While I definitely can't count how often I've seen elephants on the roads/ in the open areas in Khao Yai. I think the encounters in the forest are less than 5% of all sightings.
The road heading to Haew Narok waterfall is probably the best for elephants. Especially around the two saltlicks. Though, they are seen throughout the whole park. A late afternoon drive could be successfull. December - February elephants seem to be seen a bit more frequently.
Gaur is not so easy, though if you visit in the months April - May they can be found almost daily in the more quiet grasslands (Usually after 17:30). In the wet season I haven't had much luck with Gaur. Tracks show they seem to spread more through the forest and seem to be less dependent on the open grasslands.
Though Kui Buri might offer a better chance for both elephant and gaur, but I can't confirm that; haven't been there yet.
Gibbons (especially White-handed Gibbon) and Black Giant squirrels are often seen on the hikes and even in the taller forest along the roads (especially the stretch between the first viewpoint (on Pak Chong side) and the visitor center. A local guide will definitely make a big difference as they know the gibbon territories, fruiting trees, sounds of the jungle. But if you're like me, finding the animals is half the fun. If somebody else points it out to you, it's still great to see the animal, but it can't beat your own findings.
Don't neccessarily avoid the popular trails. E.g. for gibbons, if you stick to trails that are often visited by people, the gibbon family groups are quite used to seeing people. They are not tame as the macaques on the roads, but at least they might give you a chance to take a picture. The deeper you go in the forest the more shy the animals are. They will often be gone before you can actually see them. And all wildlife including the more exciting stuff like bears, elephants, and even some of the cats are seen on even the most popular trails.
Sambar, Indian Muntjac, and Northern Pig-tail Macaques are quaranteed in Khao Yai, but therefore not the most exciting. Though I always enjoy it if I encounter any of these species on a trail in the forest. Actually I might have seen more elephants on the trails than Sambar deer, haha...
Malayan Porcupine, Small Indian Civet and Asian Palm Civet are often seen on the night safari (though you need luck to get a good (slow) driver and a dedicated spotlighter). Costs are 500 baht for a private car, which is definitely a better option than the 50 baht join-in truck that often doesn't seem to stop for anything small like a civet...
Visit for as many days as you can. Time is the key to success!
Good luck!