since you are new to AZ Im gonna think you are looking for variety, a hike that has gorgeous views, domes, cliffs, and valleys. There are a bunch of hikes that just mosey through Ponderosa pines, PUDs (pointless ups and downs) pines and more pines. I dont think there are any hikes that can show the complete variable terrain of our state, but here are 2 that will give you a forest on top and a riparian campsite amongst a high desert-shrub area. Of course, these are only off the top of my head and others might not agree that these are worthy, but I liked 'em.
Seasonal for the month of August, of course.
1) Highline trail from Washington Park east to Tonto fish hatchery. Expansive and wonderful views especially at sunrise. 20 minutes northeast of Payson
2) Sycamore canyon via Kelsey trail to Geronimo Springs. Short, but steeeeep! 30 minutes southwest of Flagstaff.
i'm not really a squirrel