One good shape: Saturday morning in a downtown (Pleasanton, Danville, Alameda, or Oakland), afternoon on a trail, dinner back in town. Sunday: the ferry or the bridge path, or a long drive to the Delta or the Livermore vines if you want a landscape change.
The Golden Gate is a fine Sunday if you plan it as a trip. The Bay Bridge path is the version you can do before lunch and still have the day.
Towns
Inner East Bay
Oakland
The bridge lands here, and so does dinner.
OpenInner East Bay
Alameda
An island of Victorians, beaches, and a main street that still works.
OpenSan Ramon Valley
Danville
A downtown main street with the mountain behind it.
OpenTri-Valley
Livermore
Wine country, a science town, and a downtown coming into its own.
OpenWest Contra Costa
Port Costa
A handful of streets and a historic grain-warehouse bar.
OpenBerkeley & the north hills
Berkeley
Campus, canyons, and a food culture that predates the trend.
Open