Weather and Dress
June in the Bay Area is generally sunny and warm, unless you're in San Francisco, where all bets are off. Humidity is very low. The temperature drops off abruptly when the sun goes down, so evenings get fairly cool -- often in the high 40's or low 50's -- so be sure to bring a jacket or a sweater for going out after dark. There are almost no night bugs like mosquitos because they are controlled by sprays.
Most of the Bay Area is fairly casual, though if you plan on going to a really nice restaurant, you will want to wear either a nice sweater and trousers (for men) or a skirt (for women). Yes, it really is that casual. But you won't be overdressed if you feel more comfortable with getting more dressed up. The waiters will love you and bless you for not looking like another vagabond millionaire.
For the Wedding
The mood we're going for is dressy. It's a lunch wedding, so there's no need for tuxedos or full black-tie regalia, but suits and dresses are definitely appropriate. Make sure the kids wear something that can go on the playground without making you feel faint.