Hawes- This is like the valleys Disneyland for mountain bikers. A lot of love has gone into Hawes lately and it gets ridden a lot, the brake bumps and blown out everything says so. Hawes I would say are my home trails and ride them every week. I will say that Red Mountain Rush (RMR) is my favorite trail on the mountain, second after that is definitely Gidro. Red Mountain rush is reached via two ways, the better of the two would be Aloe de huez. RMR is a flowy downhill trail, with few spots to get your tires off the ground and entirely downhill but one section requires 30 seconds uphill. There are no real technical features, just an all-out scenic delight. once you're done with RMR you can go session the rest of the trails. My second favorite trail would be Gidro, more technical but still a flowy awesome trail.
Gold Canyon- If you like getting away from it all and don't mind not having long descents Gold Canyon is my second favorite place to ride. The trails are plenty and varied. My favorite is the K trail, among many others such as Phantom and Quartz Ridge. Bring your cactus combs and expect to see very few people and plenty of hours of trails while riding up against the majestic superstitions.
Ride sample https://www.strava.com/activities/9782015470
Flagstaff- Where do I begin, this is my favorite spot to ride in the state! I have only had 5 or 6 rides in Flagstaff but each one of them always left me extremely satisfied! They have been building actively for the last few years adding bits and pieces to the network. My favorite trail, hard to choose is the Heart Trail. It takes off from the top of little Elden and flows down the mountain 1,500 ft of flowy narrow scenic trail. More favorites such as big bang, a 1-year-old fun trail, Fort Valley/AZT, moto, and heart. Most action starts at the south Shultz parking lot.
Payson- There are a few great trails to enjoy while in the Payson/Pine area. I will name a few of them and at the top of that list is the newly renovated highline trails starting from the Pine trailhead. This section of the AZT is an absolute delight to run as an out-and-back or make a loop with a forest service road return. The best section is 19ish miles one way and is a stellar bit of scenic pleasant singletrack.
Another great ride is the strawberry trail that starts in Pine and can continue onto Barefoot and Pine Canyon trail. All very good single-track and makes for a great loop.