Normally, we like to take a "tour" of a city to get our bearings as much as to get to know the sights. Below is our recommendation on the best way to do this on a budget.
The first thing you should do is get one of the transportation passes if you're staying for 2 or 3 days. Pass up on the single ticket as it'll save you time and trouble—you can get one from the ticket machines but make sure you stop at the tourist office to pick up your book of vouchers if you get a WelcomeCard and at the minimum, the pamphlet Discovering Berlin by Train and Bus. This will tell you about the all important buses 100 and 200 (the only public transportation we know of in the world that could double as a tour bus) and the sights that you'll drive pass.
Start out at the zoo (Zoologischer Garten), take bus 100 to Alexanderplatz and then bus 200 back. Start at the terminus to get a prime seat at the front of the top deck. You'll pass the shell of the Kaiser Wilhelm Gedachtniskerke church, the Elephant Gate of the zoo, the Siegessaule victory column, the Schloss Bellevue residence of the president, the House of Cultures, and the Reichstag. Along Unter den Linden avenue, you'll pass the state library, Humboldt University where Einstein taught, Lustgarden and museum island, the Berlin Cathedral (Dom), the Rotes Rathaus townhall, and end at the TV tower, just to name a "phew".
Afterwards, take the S-Banh to Unter der Linden and take a free New Berlin walking tour to visit the other unmissable tourist spots: the Jewish Memorial, Checkpoint Charlie, sections of the wall, Babelplatz, Potsdamer platz. Finally, stop in at the Sony Center for some entertainment or at KaDeWe for some shopping. Most guidebooks will give you more info than we can do justice to here.
Climb the dome of the Reichstag for fantastic views of the city for free (instead of the TV tower). Be prepared for long lines though!
Give the Wall Museum at Checkpoint Charlie a miss which we felt to be highly overrated and expensive, i.e., a typical tourist trap.