Lufthansa is quite alright, and the obvious choice for flying to Germany. United's got a big presence in Frankfurt. If you opt for Frankfurt as your German gateway, there are excellent connections on the ICE (high-speed train) to all of Germany, as well as the BENELUX countries and France. The train station is at the doorstep of the airport. The city of Frankfurt itself is pretty boring. Suggest you venture off towards the Rhine or Mosel. If you're going in the winter time, the Alps is but a short 2-4 hour drive (depending on traffic, the car you're driving and your willingness and ability to go really fast) from Frankfurt, is stunningly beautiful and offers great skiing. The old parts of Munich is also very nice, and is sort of on the way to the Alps. All of the Western European countries (except Switzerland and the UK) are in the Schengen area, so you can wizz over any border without stopping for costums or immigration checks.
Car rental is another option for getting around, but if you've never been to Europe before the traffic culture may come as a bit of a shock. For one, they'll drive a lot faster than what you're used to, always have one eye in the rear view mirror. No matter how fast you go, there'll be someone in a something way faster overtaking you. Setting the cruise control to 200+ km/h is not uncommon on those parts of the German Autobahn where there are no speed restrictions.
One word: Grolsch!
While certain parts of the 'Dam may be funnier than Germany to some people, Germany still has plenty to offer. And if you're into that kind of thing, you can always go to Hamburg and visit the Reperbahn area.
PS
4-7 days is not nearly enough to see even a fraction of Germany.
