In reply to pheller :
The Apache Trail to Roosevelt Lake has been closed since 2019 due to rockfall. There are plans to clear it by 2025 but I haven't seen anything set in stone quite yet.
That said, the drive down AZ88 to Tortilla Flat and up to pavement's end (where Apache Trail starts) is a great one. It's very popular with motorcyclists and Miatas.
Flatiron is a pretty hike but it is not a short one! It's been years since I've done that one: It kicked my butt!
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If it's unique nature you're after and you have dietary restrictions, the obvious answer is Sedona. It's a couple hours north of town and has world-famous rock formations and colors; it's truly a beautiful place. It's also heavy on the hippie influence and you will be able to find food to suit your diet. If you have a decent SUV you can drive up Schnebly Hill road, which is a neat run; this is the hill that the Pink Jeep tours operate. I did it in my stock Jeep Cherokee Trailhawk, though I did use the rear locker once near the top. If you're not driving, well, you can book a seat on one of the pink Jeeps easily enough. In November it will be chilly, though.
I do stand by my recommendation of the Natural Bridge as well if you're looking for interesting nature. There are also javelina at the top quite regularly. If you keep going north on 87 you'll go through Payson, Pine, and Strawberry. The Pie Bar is a phenomenal empanada shop off the side of the road up there but I don't know how they do with low-fat/low-salt.
If you want a twisty drive in town with a neat view at the top, give South Mountain a go and drive to the lookout. If you want a shorter, moderate hike then Camelback Mountain is very accessible.
The Pima Air and Space Museum is fantastic (and huge!) and is less than 2 hours away, just outside of Tucson. If you're down that way the Titan Missile Museum is worth a look. These are museums in that they preserve artifacts (e.g. the SR71, a Titan missile) but are not exactly curated assemblages of art.
If you're recreational shooters there are firearms rental and gun ranges aplenty.
Lake Mary is gorgeous but the area up near Flag will be pretty chilly in November, likely. All the lakes near Flagstaff are great spots.
Old Town Scottsdale is okay if you're into shopping. It's not really my scene but if you're in the neighborhood of Scottsdale Civic Center, take a look at the new amphitheater there. I had a hand in building that.