One of the difficulties that I found in visiting Berlin is that everything is spread out so far — unlike most European cities where there is a “center” to begin your exploring, in Berlin there is no one central place; it was in fact two cities for many decades. Therefore getting yourself oriented and feeling like you get to see everything you want to can be tough, but I think there are two good ways to do this, get an overview of the city.
One is to take one of the (admittedly touristy) boats along the river, for which there are many different options and points of departure depending on how much time you want to spend, and I think this was really worth doing because it takes you past (and gives you a better view than you can from the street) of a ton of the city’s most important sights — Museen Insel (Museum Island), The Reichstag (Parliament building), the Chancellor’s house, the memorial point for people who died trying to swim from the East to the West and so on (yes, some of these sights are decidedly more cheerful than others…).
The other way is a cheap, open secret in Berlin: there is a city bus line, number 100, which runs from the East part of the city by Alexanderplatz all the way through many key central neighborhoods to the West, finishing up not far from the beautiful Tiergarten (Berlin’s “Central Park”). For a simple city bus fare you can get basically the equivalent service of those hop-on-hop-off tourist buses. Your hotel or even people in many shops can tell you easily the nearest place to get the 100.
And finally, if I could recommend only one sight to see in Berlin, it would be the Pergamon museum on Museum Island. Having grown up seeing classical art in abundance at the Met Museum in New York, I can say that the Pergamon museum’s collection rivals all the world’s top museums — it has not only a huge Greek and Roman art section, including the entire facade of a Greek temple, but also extensive, unparalleled collections from the East, from civilizations in what is now Turkey and thereabouts. Absolutely worth seeing.
For good restaurants I can recommend walking the area near the opera house, the university, and down towards the Brandenburg Gate — I particularly like Dresner’s which is on Unter den Linden, for very good quality German-European cuisine, and nice service.