The John Muir Trail is 211 miles and runs from Yosemite to Mt Whitney near Lone Pine, CA. It is a great place to go for a remote wilderness experience and for cooler hiking temperatures due to elevation.
The southern portion is 93 miles long and can be done in a week. There are several lateral trails along the JMT that lead to the main trail so 3-4 day excursions are also possible. One popular entry point is from the South Lake trailhead near Bishop, CA. A back country permit can be obtained from the White Mountain Ranger Station in Bishop. The permits need to be reserved 6 months in advance of any hike but there are also a set amount of walk-in permits available. Check the national parks website for more information.
From the South Lake trailhead the hike goes uphill for 6 miles over Bishop Pass. From there it is another 6 miles downhill to the LeConte Ranger station and to the intersection with the JMT.
From LeConte, head south for a week. Mather, Pinchot, Glen, Forester and Trail Crest at Mt Whitney are all mountain passes that will be crossed along the way. Be sure to train for this type of hike by getting into elevation for mutli-day excursions beforehand. The Sierra Club recommends 6 peaks to summit before trying Mt Whitney which would also work as training for hiking the JMT. Also be sure to research back country preparedness to know what food, clothes, shelter, and emergency equipment to bring along.
The John Muir Trail offers a respite from the craziness of our modern busy lives and an opportunity to enjoy an unparalleled adventure in the wild.