Eventually, Berlin will open their shiny new airport, Berlin Brandenburg Airport, which is down in the south east of the city (next to Schönefeld). When it does open, transport should be easier.
For now, Tegel Airport is fairly well connected by public transport, but only by using a bus to get to the S-Bahn or U-Bahn. Tegel Airport is located just south of the town of Tegel, which is to the north of the city. Tegel (town) is well served transport wise, having both the U6 U-Bahn and S25 S-Bahn. The stops (Tegel / Alt-Tegel) are close together (5 minute walk), but they're both in the town. It's a short walk from there to the north edge of the airport. It's a very long way from there all the way round the airport perimeter to the terminal, which is on the south.... If you have made the mistake of ending up in Tegel the town when aiming for the airport, it's about half an hour to correct your mistake. Take the U6 south (entrance is a short walk, sign posted from the S25 station, and you can even pop into the amazing Tegel Market that you pass on your way, if you're not really late!). Get off at Kurt-Schumacher-Platz station, walk up to the surface following the signs for the 128 bus. Take the 128 bus to the airport, takes about 10 mintues.
In your case, you'll be starting at the airport, so there's not that problem (well, until your return!). There are 3 buses that leave from Tegel Airport to link you to the rest of the transport network. Bus 128 goes to the U-Bahn at Kurt-Schumacher-Platz (ideal for the northern suburbs), the TXL bus goes into the city via a few stations, and the X9/109 goes to Zoologischer Garten via Jakob-Kaiser-Platz (good for the west of the city). There is a BVG kiosk in the airport, near the exit to the buses. You can buy tickets there (including multi-day passes, which you might well want), and they can give you maps and advice. Expect a short queue though, it's popular!
I'd suggest you take a look at the TXL bus page which describes the route and times, then download the BVG train+select bus map (downloads are in the bottom right of the page). Most likely, the TXL bus will be your best bet, but it's worth exploring the other options too briefly, especially if you might be returning from elsewhere in the city.
Finally, don't forget you can use the BVG Journey Planner to look up times, routes and fares for your route. Type in the airport and the address where you want to get to, tell about 30 minutes after your flight lands, and see what it recommends!