Mount Price is the second highest peak in Lake Tahoe's Desolation Wilderness. Despite the frequent crowds in the Desolation Wilderness, Mount Price is fairly remote (by Tahoe standards) and therefore sees relatively few ascents compared to the other popular area peaks like Mount Tallac, Dicks Peak and Pyramid Peak.
Like its southern neighbor Pyramid Peak, Mount Price sits atop the Crystal Range, a subsidiary crest that lies just west of (and much higher than) the main Sierra crest. The Crystal Range is a beautiful sight when seen from anywhere in the Desolation Wilderness, and Mount Price is part of the "big three" in the highest part of the range (consisting of Pyramid, Agassiz and Price). A high traverse of these three peaks is a challenging and very rewarding adventure.
From the West. The easiest approach is from the west, at either the Lyons Creek Trailhead (for the SW Face route) or at the Twin Lakes Trailhead near Wrights Lake (for the NW Face route). Note that the west side access is summer access only.
For the Lyons Creek Trailhead, take US 50 to approximately four miles east of Kyburz, and turn left on the Wrights Lake Road. Follow this road north for about 4 miles until you come upon the "Lyons Creek Trailhead". This is the same trailhead used to access the west ridge of Pyramid Peak. Follow the Lyons Creek Trail for about 4.5 miles to a fork. The right hand fork goes to Lake Sylvia and the west ridge of Pyramid. The left hand fork goes to Lyons Lake.
For the Twin Lakes Trailhead, there are two ways to get there. The first way is from Highway 50 approximately four miles east of Kyburz, turn left on the Wrights Lake Road and continue north for eight miles to the Wrights Lake Campground. At the campground take the right fork for the Twin Lakes Parking Area. The second way is from Highway 50 approximately ten miles east of Pollock Pines; turn left on Ice House Road. Continue north for eleven miles and turn right on the Ice House/Wrights Lake Tie Road. Follow this road for approximately eight miles and turn left on the Wrights Lake Road. The campground will be two miles north of this junction.
From the East. Approaching from the Desolation Valley area is difficult, as the east side of the Crystal Range drops precariously down to Lake Aloha. There is a relatively benign route, however, that ascends the granite slabs leading up from Mosquito Pass (above Lake Aloha).
From the South. You can also approach the peak from the south on a high traverse of the Crystal Range, which would require climbing over Pyramid Peak and Mount Agassiz, and then continuing along the ridgeline of the Crystal Range.
Permits are required year round in the Desolation Wilderness, one of the most popular "wilderness" areas in the US. There is currently no quota for day use, and permits can be self-issued at most trailheads into the wilderness. There is, however, a rigid quota system for overnight use, a $5/night fee (max $10), strict "zone" restrictions on where you can camp on your first night in the area, limits on group size, etc. etc. Requirements change frequently, so check the USFS Desolation Wilderness page for the latest info.
Agassiz can be climbed anytime of the year, although access to the west side trailheads is snowed in during the winter. Winter access is easiest from Desolation Valley. Since Mount Price is pretty far from the nearest winter access points (Twin Bridges area or Echo Lakes SnoPark), you're looking at a trip of at least two days to make a winter ascent worthwhile.
Camping is allowed in the Desolation Wilderness area, but it is subject to strict quotas, as well as designated "zone" requirements. See the information linked to in "red tape" section above.