I was born in Munich and spent many years of my life there. As far as the rest of Germany is concerned, I know it less well than many other countries. I guess that's because from a cultural and linguistic point of view, it always seemed a bit boring to me to go traveling in Germany. Also, if you are into wilderness areas, densely populated industrialized Germany may not exactly be the country of choice... Coming from the south of Germany, neighbouring countries to the south or southeast such as Austria, Italy and France were really close and much more appealing to me than Northern Germany. I mean why would you bother going up to the windy and cold North Sea if you are roughly the same distance away from the Mediterranean? However, I admit that these are highly personal views that not all of my German friends will share :-).
As far as cities are concerned, there are three great cities in Germany: Munich, Hamburg and Berlin. I would rank them in this (descending) order of preference when it comes to the best place to live in Germany. However, from a touristic point of view, if you can only see one city in Germany, it will obviously have to be Berlin.
Of the top three German cities, Munich has the best climate, the best strategic location (if you like to travel to more southern countries), and by far the best outdoor opportunities, with the Alps just a couple of hours away. It is green, clean and incredibly safe for a city that size. Munich is also a very wealthy city, with many nice buildings and houses. Octoberfest (in the last two weeks of September) as well as the many traditional beergardens in the summer are some typical Munich features. The downside is that Munich is a very conservative city that can feel a bit like a big village at times. I wouldn't exactly characterize Munich as a bustling big city with a hip alternative young subculture...
Englischer Garten
The Englischer Garten is my favourite part of Munich. You won't find a park like this in any other city (at least I haven't managed so far). It is an absolutely huge and very natural park, hardly ever interrupted by roads. It really is a great escape from the city where you can completely get away from traffic noise and also from other people (if you venture into the quieter northern parts of the park). In the busier parts of the park near the University and the Münchner Freiheit, it is also a popular place to play football or to sit outside and have a beer. The area near Monopteros (close to the University) is the most alternative and free spirited part of the park, with drumming sessions and lots of nude sunbathing (Americans, be warned :-)). The Englischer Garten also connects to the Isar (the river that runs through Munich), another great area for walking and cycling and for sunbathing, swimming and BBQs in the summer.
City center
The area around Marienplatz is generally considered the center of the city, with many of the well-known tourist sites. The main pedestrian shopping zone connects Marienplatz with Karlsplatz (Stachus). There is also a really nice traditional food market near Marienplatz called Viktualienmarkt (highly recommended). Sendlinger Strasse, another popular shopping street, connects Marienplatz and Sendlinger Tor. On the opposite side of Marienplatz, another smaller pedestrian zone takes you to Odeonsplatz and to a very nice park, the Hofgarten (that connects to the Englischer Garten, but has a completely different feel to it, as it is a more French-style, carefully landscaped garden).
Best neighborhoods for living and hanging out
Schwabing, the university district, is a great area (basically on both sides of Ludwigstrasse/Leopoldstrasse between the metro stations Universität and Münchner Freiheit. Another really nice area is near Isartor and Rosenheimerplatz (by the way, there is a great Roman-Irish-Finish sauna inside a beautiful old art deco building called Müllersches Volksbad in this part of the city). Also really popular are the areas near Sendlinger Tor and Goetheplatz.
Near Munich
Starnberger See is a very large and beautiful lake near Munich. The walk along the lake from Tutzing to Possenhofen is very nice, and there are some great spots for sunbathing and swimming on the way.
Munich's proximity to the mountains is one of its strongest assets. There are many great hikes only a couple of hours away from Munich (often accessible by train or by bus). One example of a popular mountain with a great view is the Rotwand. From the train station in Bayrischzell, take the bus or drive to the parking ground for Sillberg-Haus. From Sillberg-Haus, there is a nice trail via Soinsee to the top of Rotwand. For a more private (i.e. less crowded) summit experience, walk from the Rotwandhaus to the nearby Auerspitze which has an equally great view of some of the tallest mountains in the Alps (actually located across the border in Austria). Walk back down to Sillberg-Haus via Wirths-Alm or, if you are running out of time and don't have you car parked at Sillberg-Haus, walk over to the Taubensteinbahn and take the funicular down to Spitzingsee (there is a bus from Spitzingsee to the train station in Fischhausen-Neuhaus).
If you are up for a several day hike in a real alpine wilderness, head for the Wilder Kaiser or the Karwendelgebirge just across the border in Austria.
Cheap eateries
Near the university: The best place for cheap and tasty Asian food is a cute little take-away restaurant inside Amalienpassage. Another good and cheap place for lunch on weekdays during the semester is the student cafeteria inside the Kunstakademie (at the end of Amalienstrasse). The Italian wine place at the corner of Amalienstrasse and Theresienstrasse is supposed to have some great cheap pasta dishes. You can also get a huge takeaway pizza from Mario (actually a more upscale restaurant) and take it to Englischer Garten. If none of this works out, there is always Atzinger in Amalienstrasse that has a variety of affordable dishes as well as some fancier options and is also a popular place to have a dring among students, both during the day and in the evening.
In the city center: A good and cheap place to eat is the Asian take-away restaurant at Rindermarkt where you can also sit outside if the weather is nice.
Bars and restaurants
Just a few places that come to my mind spontaneously (far from being complete): La Boheme in Türkenstrasse, Blue Nile (Ethiopean food) near Münchner Freiheit and Wirtshaus zu Brezen (Bavarian food) in Leopoldstrasse. I will add some more places to this list soon...
Hamburg is probably the nicest city in Germany (if you ignore criteria like climate and outdoor opportunities). It truly feels like a big city and there are a variety of neighborhoods that all have their distinct character. The city center area around the Binnenalster is very beautiful. Further up along the eastern side going towards the Aussenalster is an extremely wealthy residential neighborhood with amazing houses. A short boat trip or train ride away is Blankenese, one of the richest neighborhoods in Hamburg with many beautiful houses built in a terraced fashion on a hillside overlooking the Elbe. Walking up all theses steps from the river to the top of the hill and taking in the view is great fun, particularly if the weather is nice. However, by far my favourite neighborhood (where I lived for almost three months) is the Schanzenviertel near the metro station Sternschanze. An ideal neighborhood for me, with beautiful art deco buildings, nice courtyards, lots of ethnic restaurants and takeaways, many shops including several great Turkish grocery stores, trendy and alternative bars and clubs, an alternative cultural center inside a squat, cinemas, theatres, you name it!
Germany's most interesting city from a historical and touristic point of view. Also by far the trendiest city in Germany, with all kinds of subcultures and an outstanding nightlife. One of the best areas to live in Berlin in Prenzlauer Berg. However, as a tourist, you will probably start by visiting the government district (don't forget to visit the Reichstag and walk up the stairs on the inside of the large dome), the famous avenue Unter den Linden, Checkpoint Charlie etc.. Tiergarten is a nice city park. Berlin also has quite a number of pretty run-down, desolate, grey-looking areas. The city is much poorer than Munich or Hamburg. There are also relatively few historical buildings or nice old houses in Berlin. Nevertheless, there is something unique about this city that maybe comes with the resilient spirit that Berliners developed during the long years of the Cold War.
There are many pretty small towns in Germany such as Heidelberg or Marburg. Germany also has some nice national parks, including the Wattenmeer area along the North Sea (by the way, in case you are in the area, take a boat to Langeoog, a very pretty car-free island), the Bayerischer Wald (that forms one large old-growth forest area with Sumava National Park in the Czech Republic), and of course the Alps, particularly the Berchtesgaden National Park. For more information on German national parks, see the German federal conservation agency website (in German).