A great outdoorsy place to visit is Muir Woods. Absolutely beautiful park about 30 mins north of San Francisco. Are you going to have a car when you are there?
Visit San Francisco. Alcatraz is actually pretty cool to visit. There are usually a few outdoor concerts/festivals at Golden Gate park you can visit. Oakland and San Francisco have tons a of different concerts and nice venues. Google around and you can find something you like. I wouldn't go to EDC unless you wanna roll. Its just a bunch of kids on e. SF Giants have a great park, worth it if you like baseball.
I would also tell you to go to a Sharks games but that's if there is a season. Anyway you would be there during the summer. San Jose has a MLS team which is actually pretty popular locally (as far as soccer goes). There are supposed to be getting a new stadium soon.
I live in Berkeley so I don't know too many local things in San Jose. They have a ton a dispensaries if you are into that. You can find places that don't card. San Jose has a nice little downtown but it really doesn't have that much compared to SF and Oakland.
If you can make longer/farther trips: Like #21 said, Hearst Castle is boss. Also consider going to Napa/wine country. The Tahoe/Sierra's are also beautiful in the summer. Go to Pismo Beach.