Mount Whitney is universally famous as the highest mountain in the United States outside of Alaska, as well as the highest point in California and the Sierra Nevada. Few major peaks have a more pronounced Jekyll-and-Hyde personality; the east face features sheer cliffs plunging thousands of feet down to the Owens Valley, while the west side provides a much more gentle aspect. The summit area is very flat and broad, one of the more anticlimactic summits of a famous peak anywhere.
Even though the east face of Whitney is the steeper, cliffy side of the mountain, it overlooks U.S. 395 and provides the easiest and quickest ways to the summit. Access from the west is a multi-day backpacking trip across the great gash of the Kern River canyon from the Sequoia National Park roads.
By far the easiset route is the Mount Whitney Trail from the Whitney Portal Campground at the end of a paved road from Lone Pine. It is a graded, well-maintained trail that uses hundreds and hundreds of switchbacks to gain the crest of the Sierra a mile or so south of the summit, and then winds along behind the craggy pinnacles of the crest to the main summit. The main obstacle is bureaucracy; the climb is so popular that camping reservations often need to be made months in advance. One strategy for avoiding the hassle is to make the trip a very long (almost superhuman) day-hike, avoiding the need for a camping permit.
There are many other routes, including the "Mountaineer's Route" on the east face, a nice class 2/class 3 alternative to the throngs on the trail. There is no shortage of world-class big wall rock climbing on much of the east face.