The whole city is very well connected via public transport be it bus, tram, U-Bahn (subway/underground), S-Bahn (suburban trains), bike lanes and boats. Make sure you buy a travelcard valid for your whole stay. Remember that you must validate the card before boarding. The BVG app is a good tool to have on your mobile.
Renting a bike is probably the best way to fully enjoy Berlin. The city is flat, there are bikers everywhere (even on sidewalks) and you don’t need helmets (unless you want to be on the safe side). The trams have special carriages for bikers.
Bus lines 100 and 200 are great for sightseeing, especially if you are lucky enough to board a double-decker and sit right in front, on the upper level.
Uber works and it is worth downloading a good taxi app as well.
The quickest way to cross the city is via S-Bahn. All of the metropolitan lines going East-West do the same itinerary in central Berlin, from Ostbahnhof to Savignyplatz and vice-versa.
Once upon a time, Berlin had three airports. Tempelhof was turned into a park because it was too central. Tegel (TXL) is the busiest one nowadays, hardly coping with added traffic and holding there just because the brand new Berlin Brandenburg (BER) airport has not opened yet. Schönefeld (SXF) is also busy, very close to the new airport and a hub for low-cost airlines. From Tegel, you get into Mitte in less than 20 minutes. There is a bus going all the way to Alexanderplatz. A taxi ride will not break the bank. From Schönefeld the best way is to take the S-Bahn until you are in central Berlin, then change to a bus, taxi or U-Bahn.