From the western end of the small town of Mancos, follow CR 42 north until it enters the national forest and becomes FR 561. After the Aspen Guard Station, go right on FR 350. Keep right at subsequent junctions (there are two of them) until you reach the signed Sharkstooth Trailhead about 19 miles from Mancos. There is good signage from Mancos. Hesperus Mountain looms above you to the right; it is brilliant at sunset. There are no real campsites at road's end, and there may be sheep around, so sleep in your SUV or arrive early. The trailhead is at 10,900'. The last mile is rough but will usually be passable to passenger cars. You might need 4WD early in the season or after a good rain.
Sharkstooth Pass is about 2 miles and 1000' from the trailhead. The second mile, through alpine tundra and flower-bejeweled meadows, provides views of the surrounding peaks, including Sharkstooth.
From the pass, head left (north) up talus and aim for where Sharkstooth's west ridge starts its climb. The rock is loose, but the going is Class 2 and pretty easy. The ridge is harder because the rock is quite loose (a good intro for people considering longer routes on higher San Juan peaks). The ridge offers some Class 3 situations that at times include respectable exposure on some very rotten rock. Just test handholds and you should be all right.
The summit is small, and there was no register that I could see. The only downside to this climb is that the good stuff-- the ridge climbing-- does not last long enough, but it is still fun and worth doing, especially when you take in the view and remember that similar San Juan views often involve sharing them with dozens of other people unless you start really early.
From the summit, a steep gully (Class 2) leads back to the talus above the pass. This way is safer than downclimbing the loose ridge is, but it is not easy on the ankles and knees or the environment. If I had to do it again, I'd take the ridge back down.