I visited Berlin in the summer of 2006 with a foreign exchange student that studied at my school in America. I spent 3 months over the summer with her and her family in Fürstenwalde/Spree. However, it is only a 20 minute train ride to Berlin, so we were frequents. Here goes nothing:
U-Bahn and S-Bahn. These are the two main subway systems, and they work like a charm...however, they fill fast, and once the doors start to shut, get in because they shut fast. They are really user-friendly once you ride them a few times. I, being a student, got a pass for the whole summer; unlimited U- and S- Bahn and Regional Express travelling within Brandenburg and Berlin, even though I was American. Ask at the local Bahnhof (train station), and they'll give you more info. (Don't worry if you don't speak German, Berliners either speak English or French; I speak English and broken German and I had NO problem) I believe last year, it was something like 35 Euros for 4 months of unlimited travel. Which is not bad at all; but there is a catch: you can only travel in the state of Brandenburg (which surrounds Berlin) and Berlin. If you are not going outside this, it is a phenominal deal! If you are frequently going to be travelling outside of Brandenburg (Poland, Dresden, Czech Rep.) then it is probably not the best way because you will have to buy seperate tickets. Just be sure to bring your student I.D.. Trains going from state to state are called "Regional trains." Student passes cover U/S Bahns, and Regional trains travelling within Brandenburg and Berlin.
Tickets aren't validated on the platform; rather on the train en route, if ever. Don't worry - if you are running late for a train, just hop on and they'll sell you a ticket once the ticket checkers come around for around a Euro more (this is called your "fine").
If you aren't a student, then, yes, there is a one-day-pass, which is somewhere around Euros 1,70. Which can get expensive. And yes, they do check for tickets - especially to/from Brandenburg and Berlin (called Regional trains). So, make sure. A lot of times on the U/S Bahns, the ticket checkers can be wearing street clothes, a lot like undercover cops. And they WILL fine you. But, before I bought the student pass, I only bought one U/S Bahn ticket, and I was never checked. It's a come-and-go thing.
They also have visitor cards, which you can pick up from your hotel. They are a nominal fee, and it gives you 3-7 days of travel on the Bahn system. Ask your hotel. Hostels generally won't have those however. If you are staying at a hostel, you can go to the Berlin tourism office and pick one up.
I would only take the Bahn system when otherwise, you'd have to walk more than 2 kilometres. It's simply not worth it; seeing the sights on street level is much more culturally educating, and well worth the exercise. I'd recommend walking most of the time; taxis are WAY too expensive.
The major Bahnhofs are, from East to West: Ostbahnhof, Alexanderplatz, Friedrichstraße, and Hauptbahnhof. They're all really accommodating, and you can get virtually anywhere within the city from them - they are great! They also have lots of restaurants, from fast food to Sushi to coffee shops, they have everything.
Again, the U/S Bahns are the way to go for longer expeditions - they are very clean, and generally on-time. I hope this helps!