Flagstaff, Arizona is a small city that serves as a gateway to many of Arizona's best natural features. It is close to the Grand Canyon, Meteor Crater, and Route 66, as well as Oak Creek Canyon and the Arizona Snowbowl Ski Resort. Each of these sites would be interesting enough to visit. All of them taken together sound like a great place to visit with your dog.
When you visit a place like the Grand Canyon, a National Park, your pet is welcome, in some areas, leashed. Of course, that makes sense because the Grand Canyon has a lot of sensitive plants and animals, many of which can pose a hazard to you and your dog. And let's face it, not every dog is well-behaved, and not every dog owner is conscientious about picking up after their animal. So there are very good reasons to make sure that your dog is carefully controlled.
Fortunately, Flagstaff has a place where your dog can just be a dog. Thorpe Park has an off-leash area, called a Bark Park, that is nicely sized at almost two acres. The park is well-fenced and well maintained. They have plenty of poop bags and trash cans, as well as benches. There are dog fountains and extra water bowls. There is agility equipment which is older but is in good condition.
Local volunteers routinely leave tennis balls and other outdoor toys for your dog to play with. There is also a small dog enclosure. Your leashed dog is welcome in the rest of the park. There are a lot of paved walking paths that wind through the park. The park has a nice playground, a soccer field with bench seating and a couple of baseball diamonds, so be aware there are likely to be children around.
The Flagstaff Dog Park is open year-round, however, in the coldest part of the year, the city turns the water fountains off, so you would be well-advised to bring extra water for your dog. Beyond that, if your pup doesn't mind the cold, she will be sure to have just as great a time in January as she will in July!