South of I-580 you have the option of a 107 mile loop. All turns, no services. Gas up and get your snacks and water as this is a demanding track that will take more than two hours. It includes Hwy 130 and Cavares Road. You can enter this loop at either Livermore or at the intersection of I-680 and E. Calavares Road. Great riding regardless of how you enter. If you are on a large touring bike (K1600GTL, Goldwing or RT, you may want to skip the Calvares Road Section as it is narrow and tight. The road is on an R1200GS, but may be a little slow with an RT simply because there are so many blind turns. The GS is a little easier to change direction mid-turn.
You can also enter this loop by taking Palomares Road south which runs into Hwy 84 and Calavares Road. This is great route and works well with Redwood as an out and back or accessing Redwood from the north after crossing the Bay Bridge.
Riding to the north via East Bay gives you access to several exciting roads. Loo Cordelia as a resting spot prior to heading to Hwy 121. Follow to Hwy 128 to 29 North. This will give you several hours of twisty riding. Hwy 29 is best taken as and out and back unless you can commit to five hours of riding. If so, check out some of the multi-day rides to the north.