Transfer from Tegel Berlin to city center

Berlin Tegel Airport (TXL) is quite a bit outside the city and there is no direct train connections there - you'll have to take the bus and change to either S- or U-Bahn (S=faster trains, longer distances between stops; U=underground, fairly tightly mashed net).

The local transport company is BVG and here you find their listing of routes etc by depature station and route planning. At the airport there are four bus routes:

• Bus X9/109/128 take you to U Kurt-Schumacher-Platz (Berlin) where you can change to the U6 underground and from that the whole network is accessible
• Bus TXL takes you to S Beusselstr. (Berlin) or S+U Brandenburger Tor (Berlin) - the former allowing quick change to the S-Bahn, the latter bringing directly to Brandenburg Gate, one of Berlin's sights.

Here is a pdf (or: alternative, central parts only) of the train network (both S- and U-Bahn) - and at the top left of the central octagon you'll find the bus connections to TXL mentioned, including their connecting train stations.

For tickets there is an integrated system covering Bus, U- and S-Bahn and prices organized by zones - TXL airport is in zone B, while the center is covered by zone A, so you'll need to get the tickets for zones A&B. Tickets can be bought at machines at the airport or from the bus driver.

As to what to see - Berlin is a big city with a huge variety of sights. For a short transit visit, I guess Brandenburg Gate and the Reichstag, and current govt buildings are a must - train stations: S Brandenburger Tor and U Bundestag will serve you well. For everything else do consult any page listing sights in Berlin and then the BVG website or Google maps for train route instructions.

Where exactly do you want to go in Berlin ? Here's the BVG, the transportation company for the Berlin area, there's a route planner on their home page

There's an airport shuttle called TXL which operates every few minutes according to their website. You can buy tickets at the airport, any other S-Bahn or U-Bahn station and even on the bus or tram (reference)