How do you put down fence posts? A brief googling says they should be 36 inches deep.

Depends. Some combine the fence posts with concrete (adding weight), some add reinforcing metal posts, some do nothing. Watching fences collapse after a strong wind storm is a common sight.

Its worth noting you can generally still drill a hole into caliche.

I use an 8 foot, iron digging bar ahead of the post hole digger.