Some good suggestions so far - I second (third?) the recommendation to ride Caltrain whenever it's going near where you're going. Very pleasant way to get around. Also, the previously mentioned Academy of Sciences; the indoor rainforest alone is worth the price of admission!
Also in Golden Gate Park, the Japanese Tea Garden is very nice. Small, but beautiful.
Japantown is cool, though it's much less well-known and much more low-key than Chinatown. The main attraction there is the Japan Center Mall, which is mostly shopping... but you said you wanted to do
some shopping, so hey. A lot of neat (and in many cases, unique) stores there, as well as some good restaurants. http://sfjapantown.org/
On the topic of food, if you're downtown and want traditional Italian-style pizza, Pazzia
is fantastic. It's a small place, so you have to either get there right before they open up for dinner (which is 6PM Mon-Sat; they are closed on Sundays) or make a reservation; after 6, it can get packed! The prices aren't too bad as quality Italian food goes. Order the Margherita pizza, you won't be disappointed. Their pastas aren't bad, but the pizza is cheaper, better, and you get more food for your money.
And also if you're in downtown and looking for lunch, and it's before 1:30PM on a weekday, seek out Cafe Algiers
for an awesome sandwich. I believe they do have some veggie options (not familiar with em because I always get either spicy turkey or BBQ chicken sandwiches!). Whatever sandwiches you get, don't be deterred if the line is out the door (that's common, but it moves decently quickly) and always go with ciabatta bread!
Lastly, it sounds like you're already planning on using public transit for at least part of the trip; unless you or anyone else with you is unable to do some walking, or you just really hate
city buses, I'd say do NOT rent a car. Traffic and parking can be an absolute nightmare (and extremely costly in the case of the parking, especially downtown). The buses are a little funky, you definitely want to stay aware of what's going on around you, but I still say it's better than dealing with all the headache that comes with driving. And moving between SF and Berkeley is pretty easy thanks to BART.
Oh, and speaking of transit: there's also the cable cars. They're obviously something of an SF icon, though I personally find them a bit overrated. Yeah, they're kind of neat, but they're also slow, uncomfortable, and cost $6 per ride. I find the historic streetcars on the Market Street F-line
to be much more interesting.