Just about anywhere in Berlin should have good access to public transportation, either bus, tram (especially in the east), U- and S-Bahn.
A popular, central area to stay is around Checkpoint Charlie. Lots of sights, cafes, shops w/in walking distance, lots of transport options. Hackescher Markt is another nice area to stay, fairly central, further north than Checkpoint Charlie.
Mitte is another area where a lot of people stay, but I feel like Mitte is more business oriented, with fewer restaurants and sights than other parts of town, and may be dead on the weekends. But still perfectly acceptable as an area to stay.
The last time I was in Berlin I stayed on the west side of the city, off Kurfurstendam. Most of what we wanted to see was on the east side of the city, so it was usually 30 minutes on the bus or U-Bahn to get where we were going.
If you look at a google map of Berlin, you should be able to zoom into areas you're interested in and look at hotels/read their reviews. A number of people on these forums like the Adina apartment hotels in Berlin.
Transport from the airport is easy by bus, regional bahn or S-Bahn. When you decide where you are staying, you can put your hotel into google maps and have it give you the best route by public transit.
A cab/rideshare to/from the airport can cost anywhere from 40-65 euros depending on where you are staying.