The TrailDex is a powerful search feature.
Here I've populated the map, with the only criteria being "peak".http://hikearizona.com/map.php?QX=1982
You can narrow it down by zooming in, or selecting any of the available options (you might want to start with upping the lower end of your elevation or mileage preferences from 0 to something harder). Changing the season preferences may help too since it can get hot here at this time of year
In the high-country, the popular summer hikes are any part of the SF Peaks, Humphreys, Weatherford, and Inner Basin. The White Mtns are more than 3 hrs from Phoenix so I won't mention those. Prescott has Spruce and Granite Mountains. Tucson has Lemmon and some others, and then there's the sky islands south and east of Tucson, some which push the 3-hr range. The Sierra Ancha are a rugged range topping out at Aztec Peak.
In the valley you have smaller peaks, some more popular or easy than others (Camelback, Piestewa, So Mtn, Pinnacle, etc.), but mostly all early morning hikes at this time of year.
Of course there's Four Peaks, but that is also a little tough at this time of year.
And I agree with your thoughts on canyons. Get to know them!
edit: as I was typing, Joe beat me to it.