Coming from Europe (or elsewhere in North America) you’ll almost certainly arrive into the Bay Area by plane. There are two good choices as far as airports go: Oakland International and San Francisco International.
The easiest way to get from Oakland Airport to Berkeley is undoubtedly by taxi, though of course it’s also the most expensive option. However, given how much you’ve spent to get here, an extra \$50 might be well worthwhile if it gets you straight to your rooms. Alternatively you can take the AirBART shuttle to Coliseum BART station - this costs about $6. From there you can get a train to Berkeley BART station for a couple bucks - easy.
The mass transit option for San Francisco takes quite a long time and involves a number of changes; I’d recommend getting an Airport Shuttle bus to pick you up directly. Using Bayporter Express, this will cost about $35 and take you wherever you want to go in Berkeley. Since you probably have no idea where you’re going, it’s prudent to fork out a little extra for the first trip with all your bags.
BART is the light rail/underground system which serves the entire Bay Area, including the line from San Francisco Airport to Berkeley. Along with the AC Transit bus system, BART is the main form of public transport in the area with multiple stops in Berkeley alone. More importantly, it offers regional connections to the likes of Oakland and San Francisco, where you’ll definitely want to spend a good amount of time!
You need to buy your ticket before boarding the train. Sadly it’s not possible to get monthly passes, but you can buy prepaid tickets for up to $60 in value, or keep recharging your clipper card as required. There are also very few to zero discounts available for students, so it’s full fares all the way.
This is the 3rd-largest public bus system in the state, serving Berkeley, Albany, El Cerrito, Oakland and Kensington. Alongside BART, it’s the most popular way to travel around the city as the bus routes go straight through the UCB campus! What makes it better than the BART system is that, as a student of University of California Berkeley, you will be given a so-called "Class Pass" which entitles you to unlimited, free use of this service!
There are extensive links between Berkeley and San Francisco, so you’ll never be short of options when travelling to the City by the Bay.
Berkeley is one of the best cities in the state for cycling. Since it’s so sunny and dry out here, I’d recommend trying to get your hands on a second hand bike as soon as you land. It will last the whole year and you’ll be able to travel when and where you want, without sitting in a crowded bus or paying for fares.