is there an order to which books I should read in order of tony hillermans books?

Answer 1

These are excellent mysteries. While there is some sense of a timeline in the lives of Leaphorn and Chee the books can be read out of order without too much trouble.

Leaphorn & Chee books by Hillerman
* The Blessing Way (1970)
* Dance Hall of the Dead (1973)
* Listening Woman (1978)
* People Of Darkness (1980)
* The Dark Wind (1982)
* The Ghostway (1984)
* Skinwalkers (1986)
* A Thief of Time (1988)
* Talking God (1989)
* Coyote Waits (1990)
* Sacred Clowns (1993)
* The Fallen Man (1996)
* The First Eagle (1998)
* Hunting Badger (1999)
* Wailing Wind (2002)
* Sinister Pig (2003)
* Skeleton Man (2004)
* The Shape Shifter (2006)

The Jim Chee Mysteries
* Three Classic Mysteries: The Blessing Way, Dance Hall of the Dead, Listening Woman (1989)
* Three Classic Mysteries: Skinwalkers, A Thief of Time, Coyote Waits (1992)
* Three Classic Mysteries: People of Darkness, The Dark Wind and The Ghostway (1993)

Answer 2

It depends. Most of Hillerman's novels will stand by themslves as complete stories, however the overall continuity flows better if you read them in their published order

Answer 3

Starting with the newest book of any writer will always prompt you to read others I like to read in order of publication date.