Although many people would recommend you to go up the Mormon church to observe the nice view of the bay. But there is a better place to get this view without barriers,
Drive up from 35th ave up to the hill,( you can exit 35th ave on freeway 580) and stop by Merritt College's parking lot, and there is the view you want!
Right north of Oakland is Berkeley, home of the prestigious university, UC Berkeley. Nice afternoon strolls along downtown in Shattuck (catch some good indie films), College Avenue for some small shops, cafe's, and ESPECIALLY need to spend an afternoon on Telegraph for a more ecletic/alternative view of the East Bay - lots to take note of, great sidewalk vendors, great used music/cd stores and cafes.
Really, there's not that much of specific tourist interest in Oakland. There is great hiking right nearby, and easy access to San Francisco and Berkeley. In Oakland, Jack London Square near the bay and Lake Merritt are nice places to stroll.
Hike in the East Bay Regional Parks
Take a BART ride under the Bay to SF
Go see the UC Berkeley campus
See Oakland's Chinatown and Jack London Square (have lunch anyplace there)
Walk through downtown (City Hall area) and see the renovation that's taking place - Go to the Housewifes Market and Ratto's Market for good, inexpensive and interesting groceries
Oakland is one of the most ethnic/culturally diverse cities in the U.S. and by and large, everyone gets along.