These are in the Gold Country around Sonora and Angels Camp. There's also Boyden Caverns in Kings Canyon, which has a little stream running through it. Up in the Cascades, there are caverns formed from Lava tubes. There are some by Mt. Shasta and Lava Beds National Monument. Shasta Caverns is up by Lake Shasta.
Farther north is Oregon caves.
There are caves in Pinnacles National Monument south of Hollister, but they are formed from fallen rocks. The Bear Valley Cave is huge and has a creek flowing through it. There are sea caves along the coast, but they're small. There are lots of tunnels and mine shafts in the Santa Cruz Mountains and at Almaden Quicksilver Park, but most of them are closed. The only caves with limestone formations that I know of are in the Sierras or the Cascades.