Which options do I have to travel in Berlin in an affordable way?

Berlin has bikes available for rent through Deutsche Bahn. The prices for casual use are €3 per year subscription and then either €1 per half hour or €15 per day (pricelist in German).

For public transport, the day card is valid from the moment of validation until 3am the following morning. This is €6.90 for Berlin AB (the whole of the city) or €7.40 for Berlin ABC (including Potsdam and Schönefeld Airport). A single ticket is €2.70, so a day card pays for itself if you're going to use public transport more than twice a day.

There are also things like Kurzstrecken (short-trip tickets, valid for up to 3 U or S Bahn stops, or 6 Bus or Tram stops), and 4-Fahrten-Tickets (buy four tickets at once for a discount), which may work out for you. For example, if you're staying at Hauptbahnhof, you could get as far as the Zoo or as far as Alexanderplatz on Kurzstrecken, in which case if you wanted to spend one day in the City-West (Zoo) and one day around Alexanderplatz you could spend €5.60 for all that travel.

You can find all this information on the BVG website (in English).

One general sightseeing recommendation is the number 100 bus, from the Zoo to Alexanderplatz. You can use a normal transport ticket on it, and it goes past most of the main sights.

In addition to Edd's answer, there is also the possibility to rent a bike at bike rental companies (link is German only) or in almost every hotel. You can recognize the hotel bike rentals easily because they have black bikes with a sign outside. Prices range from 10 to 15 Euros per day.

The public transport includes more than just the subway. There is:

• U-Bahn: subway/underground trains (only U1 and U12 are above ground)
• Tram: those trams go on the street
• S-Bahn: Highspeed commuter railway that stops all over the city
• Bus: apart from the 100 and 200 lines, those mostly go in the western part of the city, where the other options are less dense

The public transport system has a good coverage of the city and it is available 24/7. Some of the U-Bahns shut down at night, but there are replacement buses called N<something> with stations above the U stations.

While those public transport means are operated by BVG and S-Bahn Berlin separately, all tickets are valid for all of those vehicles. For example, you could buy a Tagesticket in the tram with cash next to your hotel, then go to Alexanderplatz, switch to the S-Bahn and go to Bahnhof Zoologischer Garten, grab the bus to Brandenburger Tor, take the tiny, fun U55 U-Bahn to Hauptbahnhof and the tram back home, all on the same ticket. (Please don't do that, unless you are really bored.) The same is valid for weekly and monthly tickets as well as yearly subscriptions.

Note that you must stamp tickets that you buy on machines that are outside of vehicles. There are stamping machines and tickets machines on the platforms of S-Bahn and U-Bahn. Those take cash and Maestro cards. Sometimes they accept Credit Card as well. There are cash-only ticket machines in trams and tickets bought there do not need to be stamped. They are valid immediately. In buses you can only pay cash for tickets.

The tickets listed on the BVG site for tourists are not advisable unless you are planning to go to a lot of museums. If you just want to travel once in a while, do not buy those.

If you want to not bother with ticket machines, you can use the VBB mobile ticket app. It requires some effort to set up, but once you've got that it's very easy to use and you can buy tickets on the fly when needed.

Google Maps knows the timetable of all public transport means in Berlin and is very reliable (but buses are not!).

Then there are the car sharing companies car2go and DriveNow. If you have a membership in one of these from your country (if applicable), you can easily use those vehicles in Berlin.

Lastly, taxis are not an affordable way to travel in Germany, and certainly not in Berlin.