Craigslist! Uloop is also available, and CalRentals (though in my experience, people post on CalRentals AND Craigslist, so may as well scope the free service).
east bay area apts/housing for rent classifieds - craigslist <- Will let you look for apartments in East Bay. Most likely you're going to want to limit yourself to Berkeley and Berkeley North / Hills, but if your budget is really tight and you don't mind a commute via bus or BART you can check surrounding areas as well.
Find one you like, contact the person according to the ad (and if there's no contact info, use the email address attached to the ad) with any questions you have or expressing interest in viewing the apartment, and... hopefully at some point you'll end up with a match.
Things to keep in mind:
1) You will need a co-signer if you aren't financially independent. This will most likely be your parents. Make sure to get one of them to agree with this, and to have a recent (<3 mo.) credit report on hand; when you find an apartment you like, you need to act FAST to secure it.
2) Deposits run anywhere from 1-3 months of rent. Be prepared to have at least 2x the rent ready by check by or before the move-in date (1x for the 1st month's payment, and 1x for the 1x deposit -- you'll need more, obviously, if the deposit is higher).
3) Check about laundry options. Some people forget about this. In Berkeley, most buildings have a floor with coin-op laundry machines. SOME DO NOT. The ones that do not will recommend a laundromat which is nearby (if you're lucky). Just be aware that this is how it works.
That's for your own apartment, obviously. If you don't mind sharing space and want to save money, consider looking at the room options (go to the main site and click Rooms/shared). They work a little differently in that they usually require a bit less of a deposit, no cosigner and you'll obviously be sharing space and costs. Be aware that a lot of apartments in older buildings are converted -- they used to be 1 bedroom plus living room and now they're 2 bedroom. This works out pretty well because of the way the homes are built (the living rooms are still separated from the rest of the apartments by doors and such), but it does mean there won't be a common area. This may or may not bother you.
Good luck! You'll want to start looking now even if you won't be moving up until fall. The housing market is fiercely competitive and though the occasional good thing pops up in the fall, it's quickly snapped up and often lost in the sea of ads for places which are too expensive or too far.
