Flagstaff Mountain sits behind Alta, just a little ways east on the ridge from Cardiff Pass. From Big Cottonwood Canyon, the peak is fairly tucked away. A fairly easy route that I've done multiple times climbs the peak from Alta. A longer route climbs about eight miles roundtrip, climbing over 3,000 feet to the peak from Days Fork (Spruces Campground).
The more popular route to the peak from Alta is merely a 3.5 mile roundtrip hike, gaining about 1,900 feet. Start by taking the trail to Cardiff Pass. Take the right branch of the road and follow it to an abandoned old mine on the mountain at about 9,900 feet. In summer, you'll likely find lots of wildflowers along the way. From the mine you'll have to ascend 400 feet up the steep, grassy hillside to the ridge above.
The hillside climb is the most challenging part of the climb, and once you reach the ridge a track will re-appear. Enjoy the great views back down into Cardiff Fork. Follow this east around a rocky ridge and ascend the final 200 feet to the summit. With the route exposed to the sun (south-facing), it is a good climb in June, and in early-winter when the ski resorts are starting to accumulate snow. The trip down is easy except for the steepness of the hill that must be descended. Roundtrip time should only be about three or four hours.
It should be noted that despite the easy climb, the view from Flagstaff is outstanding. I would say better than Sugarloaf/Baldy/Devils Castle offer. You'll see the rugged Superior/Cardiac ridge to the west. You'll have a partial view downcanyon, and a clear view of the Pfeifferhorn to the southwest (see picture). Across the canyon you'll have a full view of the Alta and Snowbird ski areas. A slightly higher peak sits a little further north, and getting there is not too difficult, except for a short bit of scrambling near the lower, main Flagstaff summit. From the ridgeline, you'll be seperated from Cardiff Pass by a nameless 10,320 foot peak that is quite rugged and difficult to get over.
Flagstaff is one of those peaks easy enough and pretty enough to climb every couple of years. The views, even down lower are very pretty. It would be a good tune-up for a tougher climb of nearby Mount Superior.
2002 Update: I made two ascents of Flagstaff in July 2002 within five days of each other. First, on the 20th I made a 4th trip to the peak from Alta via the standard, short but steep route that most hikers take. Nothing particularly notable other than the abundance of wildflowers out, but that's not unusual here. On the 24th I decided to climb the peak via Days Fork in Big Cottonwood, by first climbing the slightly higher "North Flagstaff" peak (elev. 10,561') and climbing the connecting ridge.
This was the first time I'd hiked up Days Fork, and it did not disappoint. As you enter the Spruces Campground, the road splits in four directions, take the middle-right road and it will take you to the trailhead (signed). The trail climbs steeply at the beginning before leveling off where a split leads to a meadow to the north (Greens Basin). After climbing gently for a while, you come to another meadow where I saw a moose drinking from the stream nearby. There were many trees here ripped up from the powerful avalanches that frequent this area. The rugged Reed and Benson ridge towers above to the west. After this meadow, the trail climbs up the right side of the valley steeply to another meadow, again full of avalanche debris. Just before reaching the final cirque basin in Days Fork, you'll arrive at the Eclipse Mine where there is a great deal of old mining equipment still present. Be sure to look back down the canyon across to the Gobblers Knob/Reynolds Peak area.
From the cirque basin neither Flagstaff or North Flagstaff stand out, but you can identify them. Flagstaff is the more rugged peak in the southwest corner, while the higher North Flagstaff is the tree-filled peak due north of it. Look for the minor ridge leading down from North Flagstaff to the cirque basin. It is a climb of about 900 feet to the summit from the end of the trail. It starts gently, but becomes quite steep as it climbs higher, and some trekking poles came in quite handy for me. There is a faint trail here, but if you lose it, just look for the route with the best footing and you should be fine. North Flagstaff has an equally outstanding view. While the view to the ski areas is not as good, you do get a view down the Reed and Benson ridge, and a view of Twin Peaks through a notch in the Superior/Cardiac Ridge that you don't get from the Flagstaff summit. Taking the ridge to the lower, main Flagstaff summit is an easy descent of about 100 feet to the saddle between. The part nearer to Flagstaff is tricky and involves a short scramble that requires some caution. It is possible to take the ridge east from Flagstaff around the Days Fork cirque to a trail that leads back to Silver Fork, then back to Days Fork. Total time for this hike was five hours.