To get a library card, you only need an address. Take some official mail with you to prove it. You pay in advance for borrowing, they put 20.- on the card,
and you can take books immediately. I think they only cost when you are overdue, then it gets expensive like 3.- a week. Videos and CDs cost. All the Berne public libraries (about 8 of them) are on the same system. The Kornhaus has the best selection, but I think all have Enlgish books. THe next book sale at St Ursula-s is on 29 and 30 June, they have about 3000 to choose from at prices from 1 to 5 francs.
Such questions are answered on swisscc, the mailing list for English speakers in Berne, http://groups.yahoo.com/group/swisscc/