The place that most people go is Aravaipa Canyon, and perhaps someday I will write some about that once I go there again. You can access it from the west (from north of Oracle and Mammoth) or from the east (near Klondike, a long drive via Wilcox).
If you are in the mood to visit the Galiuros from the east, consider Ash Creek and a possible hike up Basset Peak (7663). This is also accessed via Wilcox. Basset Peak is the highest point in the Galiuros, although Kennedy Peak, which is further north is a close second at 7549.
A number of interesting areas can be visited by way of Muleshoe Ranch, now administered by the Nature Conservancy.
Redfield Canyon and the "stone house" are at the south end of the range and can be accessed from the Mammoth to Benson "highway", aka Pomerene road.
Some roads may cross State trust land, which requires a permit for recreational use. These permits are inexpensive and now easily obtained online.
Tom's hiking pages / tom@mmto.org