Public Transportation in Berlin

Using the public transportation means would often get you faster to your destination than a cab. You can find a map of Berlin’s public transportation system here (downloads a PDF file).
There are different companies that run the system, but when you buy a ticket, it covers all companies and means of transportation: Trains, the subway, streetcars, busses, etc.
Berlin is divided into two public transportation sections: “A”, which includes all the central boroughs, and “B”, which surrounds “A” and includes all the rest. However, there are no tickets just for “A”, so you’ll need to buy a ticket for “A+B” that covers the entire city (the ticket prices mentioned below refer to such A+B tickets).
If you’re leaving the city-state of Berlin (for example when traveling to Potsdam, to Sachsenhausen, to the Schönefeld airport or from this airport to your hotel), you’ll need to buy a ticket that also includes section “C” (these tickets are valid for all three sections: "ABC").

We have an honor system in Berlin, which means that you can just walk into the trains and trams. You only need to show your valid ticket if you're asked for it, except on the bus, where you need to show it to the driver as you walk in.

When you buy a ticket, it is not yet validated, which means that you don't have to use it specifically on the day you bought it. Validating your ticket is done with the machines in the stations, and once your ticket is validated with a time stamp, you can use it. Please note that you should only validate/stamp your ticket once. Tickets bought on a bus or a tram are already validated.

 The reduced fares do not apply to seniors. They only apply to children between 6 and 14 years. Younger children don't need a ticket when accompanied by an adult (the latter, obviously, does need a ticket). Traveling with a reduced ticket when you're over 14 is just like not having a ticket at all...

 If you're caught without a valid ticket, you'll be asked to pay a fine - 60 Euro per person, in cash. Some people are surprised by the request to pay in cash and think it's a scam. Well, it's not, so you shouldn't insist on calling the police. It's your legal right to do so, but without the police, it's just a matter of civil law. Once the police is involved, it also becomes a criminal offense (fraudulent acquisition of services by non-payment").

It’s a good idea to get a day pass (“Tageskarte”) for 7 Euro per person, because a single ride (“Einzelfahrt”) costs 2.80 Euro. The day pass also allows to take with you, for free, up to three children between 6 and 14 years. It’s valid for the whole day (according to the date) until 3 am on the next day. Please note that unlike many other cities in the world, you don’t get 24 hours from the moment you bought or stamped the ticket.
If you’re three to five adults traveling together, it’s better to get the Small Group ticket (“Kleingruppenkarte”) for 19.90 Euro, which is also valid till 3 am on the next day. Small Group tickets exist only as day tickets - so there are no Small Group tickets for single rides or for more than one day.
If you’re staying for five days or longer, it makes sense to buy a 7 days ticket for 30 Euro per person (but if you're a group of five, 7 small-group tickets would still be cheaper than 5 individual seven-days tickets).
You can also get special tourist tickets like the CityTourCard and the WelcomeCard for 48 hours, 72 hours and more. These tickets cost more per day than normal tickets, but grant you discounts in many museums and other tourist attractions.

Buying these special tourist tickets makes sense if you intend to visit a lot of attractions, museums, etc. My experience tells me that most people can’t really cover much - after all, it’s your vacation and not a competition...

One of these tourist tickets is the WelcomeCard for 72 hours, which you can get, for an extra charge, in another variation: The WelcomeCard 72 hours Museum Island ticket. This ticket grants you free access (i.e. not just a discount) to the 5 museums on the Museum Island (once for each museum). This might be particularly interesting for you during summer, because having this free entry means that you won't need to stand with many other tourists in the queues for the tickets. In the winter, there are usually no queues, so you shouldn't get this special ticket unless you really intend to visit all 5 museums.
To find your way - e. g. from the airport to your hotel or from your hotel to a museum etc. - you can use Berlin's "Journey Planner", that usually delivers different possibilities for you to choose from. If you have an Android smartphone, you can download this app for free that will tell you how to get from one point to another. There's also a similar app for the iPhone right here.