Drive to the Glacier Lake Trailhead outside of Red Lodge, MT.
"This is one of the shortest approaches to any 12,000 foot peak in the Beartooth Mountains and the standard approach for Spirit Mountain. There are two variations for this approach based on which route you want to do. For the Standard South Ridge Route it's very easy. Simply hike the very well defined Glacier Lake Trail for 2 miles to Glacier Lake. This is a very popular day hike in the summer among families and fishermen.
For any other route on the peak except the North Face, use this variation which heads up to the Shelf and Moon Lakes basin. From the Rock Creek trailhead, begin up the steep switch-backing trail through an old burn area and reach a nice footbridge crossing Rock Creek after 0.6 mile at 9,080 feet. Cross the bridge and continue up the trail for eight more switchbacks and locate a side trail heading off to the right (north) at 9,230 feet, a quarter mile from the bridge. There may a cairn here to mark the turnoff but don't expect one. Follow this steep but well traveled boot path along the left (west) side of this side creek for 0.35 mile to where it crosses back to the east side of the creek on large boulders at 9,600 feet. Past this crossing the slope eases and the path continues through open grass above and on the right side of the creek. The trail can be hard to follow at times but follow it to a small tarn at 9,850 feet where it once again crosses the creek back to the west side. Continue up another couple hundred feet on open grass to Shelf Lake which makes an excellent place to camp or continue around the left (west) side of Shelf Lake and continue northwest over a large, broad grassy bench for 1.2 miles to the southwest corner of Moon Lake where you can also camp. It's about 3.2 miles from the Rock Creek Trailhead to Moon Lake with slightly over 1,500 feet of elevation gain."
-Excerpt from lemkeclimbs.com/spirit-moun…