Einsteinhaus
Albert Einstein rented this small flat with his wife during his years working at the Swiss patent office. Their first child, Hans Albert, and the special and general theories of relativity were born here, where Einstein's writing desk overlooked the busy street and its lovely clock-tower. There are numerous photos and original documents from Einstein's life, work, and speeches. It is closed until further notice due to severe water damage.
Swiss Alpine Museum
A museum showing the full variety of the Swiss mountains.
Tierpark Dählhölzli (Zoo)
Berne's zoo is located along the Aare river, with many outdoor enclosures that actually integrate the river.
Zentrum Paul Klee
The Paul Klee centre which is in a modern wave-shaped building presents the world's most important collection of works by Paul Klee rotating exhibition drawn from 4000 works, or 40% of his oeuvre. If you plan on visiting, then the CHF20 "Berne card" validated for that day show it at the ticket counter to receive a complimentary pass is totally worth its price - you'll spend about that for bus round trip and the ticket alone.
Zytglogge
The Clock Tower near the center of the old town, built around the turn of the 13th century, is a great thing to see. On the hour, every hour throughout the day, there is a stunning display of early animatronic technology. The locals are proud to tell you it's "the longest running act in show business". A few minutes before the hour, it begins with a little song and some drumming by a jester on top. On the hour, bears and an old bearded king get into the act. It's great for kids to see. The clock tells time too, as well as the month, day, sign of the zodiac and phase of the moon. There are guided tours inside the tower that will let you have a look at the clockwork while the show is displayed outside. It can be booked at the tourist office and is definitely worth it if you love mechanics.
Berne is full of history and museums. It also has quite a bit of public art, all of which is marked on a walking map which is available from the tourist office in the train station for free.
Bundeshaus (Federal Palace of Switzerland)
The Swiss House of Parliaments is a representative building dominating the Square. Constructed by the end of 19th century. Free guided tour when the Parliament is not in session. During session only access to the spectators ranks.
Invasion of Berne -- successful!
As you explore, you may notice these small alien graffiti mosaics. GAME NOT OVER was declared by the anonymous Parisian artist "Invader" in 1998. Since then, space invaders have been reappearing on the walls, bridges and roofs of cities across the world, most famously on the Hollywood sign and in several locations in the Louvre. Two additional Swiss cities have been invaded: Geneva and Lausanne. Those with 10 EUR, a longer visit, and a weird sense of humor might consider ordering a map and doing the space invader tour.
Kunstmuseum (Museum of fine Arts)
The Museum of Fine Arts Berne is known for its collection of works of painters such as Pablo Picasso, Ferdinand Hodler and Meret Oppenheim. It is the oldest art museum in Switzerland with a permanent collection and houses works covering eight centuries.