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We did some of this trip in late May / early June 2004, and although there were a few places we couldn't get to our biggest problem was finding open campgrounds. We didn't even try the "Going to the Sun Road" and most of Crater Lake was closed, but the Cascades were beautiful! One advantage of the spring is the amount of water in streams & waterfalls - later in the summer many dry up. 2004 Trip
I tried to visit Crater Lake in early April 2002 and found it closed.
The roads around Tahoe are usually kept open. Highway 50 is the main traffic artery south of the lake. It will sometimes be closed by storms, but they open it back up. They may well require chains if you go in wintertime, though.
BTW, I-80 north of Lake Tahoe is also very nice. There are some lovely scenic points of Emigrant Gap (westbound) and Donner Lake (eastbound).