Here are some of the iconic trails in the area. Imogene, BlackBear, Ophir, Yankee Boy Basin, Governors Basin and a few others that are on the West side of Red Mountain Pass or the Ouray and Telluride side of the pass.
The East side of the pass is commonly known as the Ouray, Silverton and Lake City side of Red Mountain Pass. This includes Mineral Creek, Engineer, Poughkeepsie Gultch, California pass, California Corkscrew, Hurican Pass, Cinnimon Pass and the cool ghost town know as Animas Forks. Stoney Pass is not difficult, the views are outstanding, the summit of the pass is on the Continental Divide and is where the Rio Grande River starts as a trickle from the snow pack. Not a quick drive though.
I would stay away from BackBear Pass. The switchbacks are tight, with huge drop offs, and the top section feels a little tippy. There’s a difficult section of Poughkeepsie Gulch you’d want to avoid for difficulty, called the Wall, but it does have a bypass.
Yankee Boy Basin is close. Wrights Lake is a nice destination for a day trip. Depends how fast you drive. Yankee Boy connects to Imogene Pass if you decide to have a longer drive into Telluride over a 13,000 foot pass. Amazing views !! You could have lunch in Telluride and drive the road back to Ouray.
Governor’s Basin is fairly close, off of Yankee Boy as well, with beautiful views, wildflowers, waterfall seen across the valley, and interesting rock formations at the top. One of my favorites!! Much nicer than Wright’s Lake in my opinion. This is a less commonly traveled trail, so this is where I often find solitude. Stay away from nearby Sidney Basin, which has more difficult trails.
You’ll want a mapping app. OnX is my current favorite. Easy to use. You can load the trails and route you want to drive ahead of time. Nice to use a tablet for a wider view of the map. The colors on the map below are my own selections, not showing difficulty.
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