Berkeley and Oakland combined make for a great day trip from San Francisco. Alameda and Walnut Creek are other towns worth considering if you aren't seeking the university ambiance per se.
In Berkeley, you can check out the campus and see the remnants of counter-culture along Telegraph Avenue. You can also visit downtown and the Gourmet Ghetto to get a feel for what town is like.
In Oakland, I recommend taking a look around Rockridge and Piedmont Avenue. Both are pleasant neighborhoods with an assortment of restaurants and higher-end shops. Uptown is nice for bars and nightlife. Oakland's Lake Merritt is a beautiful place to walk or jog, and the nearby Oakland Museum of California is an interesting place to visit.
Both Oakland and Berkeley are a little more relaxed than San Francisco. I'm not sure what kind of shopping you want, though. Most people buy stuff from shopping malls if they are interested in bargains. In this case, Bay Street in nearby Emeryville would be the best spot to go.
Alternatively, both Alameda and Walnut Creek may be better choices if shopping is the most important factor for you. It seems visitors from Singapore really like to shop...