The budget airlines (Ryanair, Easyjet) use Schönefeld airport, in the south-east. Take the S-Bahn into the centre of town, or the Airport Express from the same station. It’s cheap, fast, and - since this is Germany - punctual. Don’t take a taxi from Schönefeld airport to the city centre: it’s a long way.
Some airlines (like Air Berlin) arrive at Tegel airport. From here take the X9 bus to Zoologischer Garten.
Updates
As LukeHarding says, take the S-bahn from Schoenefeld, not the U-bahn. The U-bahn trains serve the airport but some are semi-fast and others stop everywhere. The S-bahn trains are the ones to go for.
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