For Cheap and Fun? 
Aside from Berkeley, Theres not much to do in the east bay. Berkeley has a lot to offer though. Lots of stuff going on around the UCB neighborhood. Fun stuff. Probably a great option for what you're looking for. If you want to live anywhere else, people either go to SF or SJ. 
Let me warn you though, driving in the bay gets old real fast. That and its really not worth driving back from a night of fun... if you can afford the option.<p>San Jose. SJ Downtown and SJSU is right next door so theres still some fun activity there too. You'll for sure drive everywhere if you live here.<p>Penninsula... Pretty much sucks if you want to do anything night life wise. Palo Alto / Stanford stuff is all there but if you're not part of "the click" it might be hard to mingle at first. PA is also expensive unless you go to East Palo Alto which is ghetto and quite unsafe. Make sure you look up where you move into before you commit.<p>SF. Awesome except for the parking part. Sell your cars and move up here! Lots to do, bikes, scooters and public transports will take you everywhere you need. Probably the best place to be in the bay. I think from everyone who posted in this thread who's ever lived here will agree.