Trail access from the east (lake side) is easy. The western side is trickier due to smaller roads and snow. There's less vertical than, say, Kings Canyon / Whitney area - you'd probably park at 6500 ft and hike up to 10,000 for the summits. The area is popular so weekends are not recommended. Overnight permits cost five buck per person, and a few of the more popular day hikes charge parking fees (Horsetail Falls & Eagle Falls).
For day hikes here's a few good ones (clockwise 'round Desolation starting from 1 o'clock (* highly recommended):
- Meeks Bay TH - Rockbound Valley, Phipps Peak.
- * Bayview TH - Eagle Falls TH loop
- * Mt. Tallac TH - Mt. Tallac
- Glen Alpine TH (or Echo Lakes TH) to Lake Aloha
- Lyons Creek TH - Pyramid Peak (tallest in Desolation)
- Wrights Lake TH - Twin Lakes (and maybe up Mt. Price)
This is probably the best source for info - the 'more information' box on the right sez it all:
http://www.fs.fed.us/r5/eldorado/recreation/wild/deso/