The lovely copper-domed building, formally opened in 1901, has served as the Capitol of both Arizona Territory and the State of Arizona State; today it serves as a four story museum dedicated to the history and culture of the state. The best way to appreciate it is on one of the free guided tours; highlights include the restored 1910 House Chamber on the 3rd floor, and the 1st floor exhibition dedicated to USS Arizona, lost at Pearl Harbor.
If you go:
Take a close look at the impressive mosaic state seal on the first floor. It was meant to represent Arizona's five C's: cattle, cotton, copper, citrus and climate but two were accidentally missed off by the contractor.
Links:

1700 West Washington Street
Phoenix, Arizona 85007-2812 USA
capmus@lib.az.us
Phone Numbers:
Info
+1 (602) 926-3620
+1 (602) 926-3620
Tours
+1 (602) 926-3628
+1 (602) 926-3628