I can't comment on Utah, but I have hiked, climbed and biked extensively in the Sierras. In the far north, the only main route is Hwy 89. This is fantastic in the area south of Tahoe. North of Tahoe, it gets mighty nasty, and I strongly urge you to avoid it.
Hwy 395 is generally excellent. My wife and I rode a big section of it from near Ridgecrest all the way to the turnoff for Monitor Pass, which was tough and one of the best roads we've ever done. Traffic varies. Note that there are some very nice options to 395 depending on your motivation and interest. Around Lee Vining/June Lake, you can head out Hwy 120 and Benton Crossing Rd to loop around the Glass Mtns--one of the great rides of the western US! (virtually no water on route, however, so carry lots) On getting up or down at Tom's Place/Sherwin summit, check a map and you'll find the Little Rock Creek Rd. which drops down through Paradise and back roads to Bishop.
Many stretches of 395 have great shoulders but there are sections along the Walker River north of Bridgeport where things get tough. These are pretty short, however.
If rock climbing is your goal, will you be carrying gear, trying to meet people, or will you be hiking? I've climbed all over the range for 30 years, so I can give lots of advice. Doing much on a bike would be tough. There is some fantastic cragging above Tom's Place and a campground at the base--Gong Show area and others--sport and traditional. Out some dirt roads east of Mammoth is Clark Canyon with loads of excellent sport climbing and free camping (surface water available--but filter it). The trick is hauling enough food and having access to water, depending on the climbing area.
As noted, getting up the east side roads to the higher altitude crags would be huge work but possible if you don't mind considerable suffering.
Note that once you get to Bishop, summer temperatures can be pretty high 90 to 100 deg. F.--not my cup of tea.
Note that excellent climbing guides are available, so do your research.
Best of luck!
Scott