transport

As I mentioned before, public transport is an important factor in Berlin. As everything is really widely spaced apart, distances which look small on the map are still a lot of ground to cover. If you check it out on google maps, calculate some distances just to get a feel of the scale of Berlin. Its HUGE.
The most famous public transport is probably the U-bahn, which covers the centre of the city. In addition to that, the S-bahn offers a connection to the less central parts, and are connected with the U-bahn in certain stations.
Map of the U- and S- bahn rails and stations.

In addition to U and S-bahns, there are buses (which travel all night long where the U and S do not) and regional trains. The website of Berlins public transport is excellent and offers a large english section.
I bought a Monatskarte, which allows me all travel on all Berlins public transport for 72 euro a month. A hefty cost, but well worth it.
Be aware that the public transport is separated in zones in Berlin; A and B being the central parts and C the outskirts of the city. A and B are the most common and will get you everywhere there's something to see.

Something which might be strange at first but is very logical once you get it is the big "ring" you see on the S-bahn network that is actually called "ring" on the information boards in the stations. Trains ride the Ring in two directions, clock-wise and counterclock-wise (this is indicated with an arrow on the information boards in the stations).
If you don't have any experience with subway systems it will be confusing at first, especially since the system in Berlin is so extensive. Having a good sight of color helps (sucks to be me) and remember to look at the endstations. If you, for instance, want to get from Ostkreuz to Alexanderplatz, you take line S3 or S75 to spandau. Often the most important stations in between are mentioned as well. Also, there are maps like you can find in the pdf file everywhere in the stations, in the S- and U-bahn carriages the next stations are announced and the BVG website offers a route planner which is a great help in the beginning.
Still be prepared for some longtime staring at maps and boards in subway stations, though :) Don't worry, all the tourists do it.

However quick the trains might ride, the S-U system still might take some time to get you somewhere, especially if you have to switch lines. Sometimes it's simply crossing the platform (and waiting a couple of minutes for the train to arrive), but if you want to get from an S to a U line or the other way around, often you have to walk several hundreds of meters and a couple flights of stairs to get to the right platform.
The system is generally considered safe, until now I only have had the typical drunken shouting of partygoers in the station, a singing lady (with a monycup) and a magazine-selling guy in the train itself. The latter were not obtrusive or annoying, however, so no problems to mention there (yet?).

Besides the public transport, you still might find yourself walking a lot, even simply going to the subway station or shop might be a long walk. This is where the other popular means of transport comes up: bicycles. Berlin is a very bicycle-friendly city: traffic is never really busy, drivers do actually pay attention to the road, lots of wide bike lanes, special traffic lights etc.. There are lots of places to park your bicycle or just put it against a wall, and a normal lock will do.
I bought a bicycle for 65 euro on Craigslist, as you can see lots of cheap bikes are offered there. I got mine from an American returning after a stay of 5 weeks, he bought it on craigslist as well. It's an excellent bike with working lights, brakes and three gears. Saved me a lot of walking already. For longer distances public transport is still advised however, because even with a bike Berlin is very big.