Berkeley and the San Francisco Bay Area (or just Bay Area) have excellent public transportation. The simplest way to get around is to use your feet, the bus (AC Transit), and the subway (BART). An EasyPass Clipper Card is included in your student fees so be sure to pick one up at the Cal1Card office. The card can be used to ride any AC Transit bus free, but it's a fully functional Clipper Card that you can load money onto. That means you can use it for Bart, SF Muni, Caltrain, and other public transportation around the bay. You can find out how to pick up your EasyPass from the Cal1Card Office.
AC Transit offers service throughout the East Bay. You can see a map of all their routes here. Select on the bus line number to see its route. Once you have chosen a bus, you can get live updates about when the next one will arrive at the station nearest you by using the Nextbus service. There's even an app, but don't bank on it being accurate.
AC Transit will take you around Berkeley and neighboring cities. BART is useful for getting to San Francisco and the two major airports: SFO and OAK. AC Transit offers a bus into San Francisco, the F bus, which is free for students. There is also an all night bus from San Francisco back to Berkeley, the 800. This bus runs every hour during the week and every half an hour at the weekends. It is not recommended that you travel on this bus alone after the BART has closed.
511.org is an excellent resource for finding AC Transit (bus) and BART (train) schedules.
Public transportation is great and all, but if you want to go to In-N-Out to get one of the finest hamburgers available or haul cinder blocks from the hardware store to your apartment, nothing beats having your own car.
A new car will probably be beyond your budget if you are on the usual stipend--at least until you get that incredibly lucrative summer internship. However, a used car can be had very inexpensively. A good starting place is craigslist. Of course you will need to get a California Driver's License and also insurance, but that's also very easy.
The most difficult part about owning a car in Berkeley is parking. It's tough enough to parallel park your car into a tiny space, but you'd be lucky to even find such a space in some parts of town (around campus particularly) at certains times of day. Not only is finding parking difficult, the city meters and restricts parking time. So pay attention to posted signage. But don't let this discourage you. Generally, parking in residential areas is easy--except for areas densely populated by undergraduates. If you must park on the street, you will definitely want to pick up a parking permit from the city. These are inexpensive and can be obtained at the City of Berkeley Permit and Parking Ticket Office. They also sell visitor permits that you can give to visitors that want to park on the street near your home for more than 2 hours.
Some students living in neighboring cities such as El Cerrito and Albany drive to campus. If you live beyond a minimum distance from campus, you can qualify for a permit to park in UC operated parking lots around campus.
Car sharing has emerged as a viable alternative to owning a car. For-profit and non-profit companies have implemented similar systems in the Bay Area and other major metropolitan cities in the U.S. where members have access to numerous cars parked at usually convenient locations for a price per hour. Since this cost usually covers gas, maintenance, parking, and insurance, car sharing is a good option for people who don't use a car very often or just want a car for a few hours to go grocery shopping.
There's a huge UC Berkeley discount with Zipcar. Membership only costs $15 for a year.