Kellog and Incinerator usually run together. It's really a one-direction trail. Incinerator Ridge is possibly the hardest trail on Mt. Lemmon and also the newest trail built by the Forest Service on the mountain.
Kellog is a very short trail in between two other trails so it makes sense to include it as one with Incinerator Ridge. It starts about mid-way on Mt. Bigelow. There's a 4-way junction at the mid-way saddle. The junction is marked with a metal sign. Kellog heads off to the southeast from here. It's techy in spots but overall there's nothing too hard. Then it comes to a dirt road, and Incinerator Ridge continues in the same direction.
Incinerator is also marked with a metal sign and is easy to follow. This trail is extremely difficult. It's loose, steep, rocky, and has very tight switchbacks. This trail will take you out to the parking lot where Green Mountain starts.