The southernmost trail up Mt Elbert is the Black Cloud Trail and is certainly one of the most beautiful trails I have ever been on. I can't recommend it enough. Even on a Saturday I only encountered two people on the way up. The wildflowers, views of the Sawatch Range to the south, streams, solitude, and varied landscape all make this a special hike.
The trailhead is especially difficult to find and easy to miss. The trail starts in pine forest and aspen groves while climbing steep switchbacks along Black Cloud Creek for about 1.7 miles and 1,700'. The route is very scenic with views of waterfalls, cascading streams, pine, aspen forests, and beautiful clearings.
Eventually the trail flattens out a little at around 11,700' before it steeply switchbacks to gain Mt Elbert's southern ridge. Once on the ridge 2.7 miles in, head north for the highest point, which is unofficially South Elbert (14,134), which is not an official 14er because the saddle between it and Mt Elbert does not provide sufficient prominence.
From South Elbert, follow the ridge northwest until you descend into the saddle. There is a faint trail crossing the saddle across the south ridge of Mt. Elbert
, keeping to the right. After traversing the ridge you'll be on the summit of Mt. Elbert.