Mount Whitney can be seen from Lone Pine in Inyo county rather easily. It actually doesn't appear to be the highest mountain from that angle, Mount Lone Pine looks bigger since it is closer. Mount Whitney can be identified as one rather angular mountain with several radically steep pinnacles to its south.
Whitney isn't the most dramatic looking mountain in the Sierras, and it isn't anywhere close to being the hardest to climb either, as there is a trail right up the side. To access this trail you have to drive to Whitney Portal, west of Lone Pine. The trail starts around 8000 feet so its really only a 6000 feet climb (still pretty hard at that altitude). The trail is about 13 miles long and has something like 100 switchbacks. The area would be rather pretty if it weren't overly infested with people. If you want a chance to legally use the trail you need to get a permit, which is rather hard to do. Personally, I'd rather climb a mountain that doesnt have a trail, is a few hundred feet lower, and hasn't had anyone set foot on it in months.
The amazing thing about the Mount Whitney area isn't really the mountain itself, but rather the sheer wall of mountain that stretches 100 miles or so north and south with no low passes or roads across it... the whole wall is about 12-14000 feet above seal level and the bottom of the valley is around 4,000 feet... leaving a sheer wall of up to 10,000 feet... and this altitude gain is all in 10 miles or less. Some of it is pretty much sheer cliff. If you want a real challenge, forget Whitney and try hiking to the crest from Shepherd or Sawmill trailheads, or try hiking up Birch Creek out of Big Pine.