A lot of people like Oakland, because it's a "real" kind of town... San Francisco and Berkeley are a little stronger on fantasy than realism. I don't know as much about the place as I'd like, but anyway, here's a few recommendations:

Some of the best avant-jazz, new/creative music in the Bay Area is happening at "21 Grand" these days, a medium size place located at 416 25th St at Broadway, Not far from the 19th Street BART station between Broadway and Telegraph. Well, the 12th street stop is only a few blocks from the Stork Club (on 12th near Webster). This is a fabulous dive of a bar, well worth a visit. They often have decent live bands play there (or at least used to when Lorrie Murray was doing the booking... I don't know what it's like these days).

If you go a half dozen blocks away from Broadway, heading south (on 10th, 11th or 12th, I don't think it matters) you'll get to the Oakland Museum, which is pretty cool by me. Some more information is down in the References on museums.

Jack London Square is in Oakland around Lake Merritt. Personally, I can't imagine why I would go to Jack London Square, except to go to Yoshi's (jazz with sushi), and I'm not often interested in Yoshi's variety of traditional jazz (though every so often they surprise me and book something really interesting). More info in the Reference.

But the one thing that really does draw me over to the East Bay these days is their creative/new/improv/unpop scene, which seems as strong as (or stronger than?) the scene in SF. The superb Beanbenders series (Reference) is no longer active, but the east bay folks (who I increasingly think of as the Mills College Mafia) are keeping the faith in a number of places. If you take a look at the calendar at the bayimproviser's site, you'll probably notice a number of east bay locations: Reference