People ask me this a lot, and I rarely have enough time to answer thoroughly (and if it's on Twitter, I don't have enough characters), so I thought it would be fun to write about some of my favorite places.
I'll start by recommending the Lonely Planet series of travel guides. They've never steered me wrong, and they have recommendations for inexpensive places to eat and sleep, which is great for the budget traveler, or pretty much anyone who thinks paying $100+ a night for a hotel room is ridiculous. There are also e-versions of them, and you can buy individual regional chapters. If you have a smartphone, this is a great way to save paper and carrying things.
Most of the popular tourist areas are in southern and west/central Germany: Bavaria (Munich, Nuremberg, the Romantic Road), Frankfurt, the Black Forest, the Rhine. So if you have a limited time in the country, you can see a lot of sights by getting off the plane in Frankfurt (second busiest airport on the Continent and a hub for many airlines) and hopping trains toward Munich and back.
I'll start with the middle: Hessen. Hessen has many rivers, including the Rhine and Main, and is very hilly throughout and mountainous in the south. Hessian Mercenaries fought in the American Revolution on the side of the British, and in my hometown there still stand Hessian Barracks, which are home to the Maryland School for the Deaf.
Frankfurt am Main is a major financial center. I've never spent much time in the city itself, just the airport and train stations. There's a shopping district, museums, and the like.
I spent my junior year in Marburg an der Lahn, a medium-sized town about an hour northish of Frankfurt by train. It's scenic, and the central old town, the Oberstadt, has some funky cool old buildings, including the oldest Gothic cathedral in Germany and the oldest Protestant university in the world. The Brothers Grimm studied at Uni-Marburg, and Rapunzel's Tower stands in the mountains nearby. The landgrave's castle high up in the Oberstadt (ober means above/over) commands an amazing view. If you have time to spare, swing by Marburg.
Fulda, in far eastern Hessen, is a name that should ring bells for anyone familiar with the Cold War: a gap in the mountains at Fulda was the proposed route for a potential Soviet/East German invasion of the west. It had been used for various invasions over the centuries, so why not?
Hessen has a lot of quaint towns that are less touristed than those in neighboring states, as well as lots of hiking in the mountains. I grew up in the shadow of the Blue Ridge, and the geography of Hessen was comforting.
Next up: Rhein-Pfalz, where I took my first trip to Germany (and barely remember).