Frankfurt Airport, take the escalator a level or two down from the arrivals level in Terminal 1 to the trains (U-bahn signs). Buy a one day pass that covers the trains and busses. Downtown (hauptbahnhof=main train station) is only 3 stops from the airport and takes about 15 minutes. Lots of stuff within walking distance, as you would expect from any world capitol.
Should you stay in the airport, there’s a few cafes on the “fifth level” that have a glass wall overlooking the airfield. At the base of the escalators that go up to the cafes there were 8 free Internet terminals. Lots of shops in the terminal on the departure level and shops, restaurants, and a grocery store on lower levels.
Munich Airport, you’re almost an hour away from Munich proper whether by cab or by train. During morning and evening rushes these times increase dramatically. Should you take the train, make sure you’re on the express into the city. Marienplatz is the place to go: lots of history, lots of shopping, lots of eating, lots of drinking. You’ll have to change trains at the hauptbahnhof to get to Marienplatz but it’s only a few stops from the hauptbahnhof. I’ve not been there but the town of Erding is right outside the airport, home of Erdinger beer.
Hamburg Airport is about 45 minutes from the airport by public transport. Unfortunately the trains don’t yet get to the airport. Take the 110 bus to the Ohlsdorf station where you can take a U-bahn train (U1) or S-bahns (surface trains) S1 or S11 to the hauptbahnhof. Again, lots of stuff to downtown.
If the waterfront is more your thing, both U-bahn and S-bahn run there. Ask at the airport’s information desk for the trains you have to take.
Berlin Airport, good luck. There are 4 airports. Try to get downtown to the TV Antenna. There may be a line but it’s worth it for the view. Sightseeing is pretty good in that area also. If that’s not an option, go to the hauptbahnhof. Within a mile’s walk, there’s a zoo, a bombed-out church, major department stores, and a Hard Rock Cafe! Next to the church is a plaza where there’s always something going on. Grab a spaetzel, a bier, and a seat for some world-class people watching.
Well, those are the only German airports I remember much about. Enjoy your time.