There are three different groups of libraries in the valley, Salt Lake City, Salt Lake County and one independent library in Murray. Here are links to each of them:

Salt Lake City Library System

Anderson-Foothill Library

Chapman Library

The City Library

Corinne & Jack Sweet Library

Day-Riverside Library

Glendale Library

Marmalade Library (Opening Fall/Winter 2015)

Sprague Library

Salt Lake County Library System

Alta Reading Room in Alta

Bingham Creek Library in West Jordan

Calvin S. Smith Library in Salt Lake City

Columbus Library in Salt Lake City

Draper Library

Herriman Library

Holladay Library in Salt Lake City

Hunter Library in West Valley City

Kearns Library

Magna Library

Millcreek Community Library

Riverton Library

Sandy Library

South Jordan Library

Taylorsville Library

Ruth Vine Tyler Library in Midvale

West Jordan Library

West Valley Library

Whitmore Library in Cottonwood Heights (Closed until autumn 2015 for remodeling)

Independent City Libraries

Murray Library