If you want to tour the famous squiggly Lombard Street (the squiggly part is only one block), I recommend walking it instead of driving it. It is nicely landscaped, and should be full of flowers this time of year. It is only about one block from the Cable Barn, so you could do both of those in one outing. (Don’t say you weren’t warned: That block of Lombard is steep, so if you walk down it, you have to walk back up!) The cable car between Hallidie Plaza (in the downtown area) and Fisherman’s Wharf has a stop there.
Re: Suggestion to view the bay from top of Campanile Tower at UCBerkeley: If you like grand views like that, you can do even better. (I assume you are driving a car.) Drive up to Grizzly Peak Road (which runs along the ridge of the “hill” on the east side of Berkeley and Oakland. Drive along this road. Stop at various turnouts. Assuming clear weather, you get a majestic view overlooking the East Bay Area. While you’re in the area, you could check out Tilden Regional Park. There’s a visitor center there, a short loop trail around a pond full of turtles, an early 1911 Herschell-Spillman carousel (article ; photo) (There’s a similar carousel in Golden Gate Park too.) (ETA: And additional miles and miles of hiking trails if you’re into that, and picnic areas.)
If you’re into outdoor recreation, in particular visiting redwoods and doing hiking, there’s [del]hills[/del] mountains full of redwoods and hiking trails all around. East Bay (along Skyline Blvd, which you get to from Grizzly Peak); North Bay in the hills west of Santa Rosa; South Bay in the hills between Silicon Valley and Santa Cruz.
For another grand view, drive over the GG Bridge into Marin County, taking note of the mountain just past the bridge on the left. At the first exit (Alexander Av, to Sausalito), get off but turn left instead of right at the end of the off-ramp. Go under the freeway (takes you towards an on-ramp right back to SF), but just before that on-ramp, turn right into Fort Cronkhite, a steep road going up that mountain. Ruins of several WWII era fortifications there; interesting geology (you can see the layers and folds in the rock where they cut into it to make that road); and a view over the Bridge, the Bay, and San Francisco like from an airplane.
Oh, and back in the East Bay: Southward, near (or at?) Union City, right at the edge of the bay, is a large marshy wildlife preserve area. (Forgot the name, but you can find it on any on-line or paper map.) Good bird viewing, probably. A small hill you can climb right at the edge of the bay, with a good view overlooking the whole South Bay region.