If there's one thing you can expect from German transport, it's that it'll run with German efficiency, so getting around the country couldn't be easier. Unfortunately it can also be pricey, and the fastest modes of transport are always likely to be the most expensive. The most popular options for seeing Germany are either by hiring a car, which gives you the freedom to see everything at your own pace, or using the extensive rail network. If these prove too expensive, it's usually possible to arrange ride-sharing in Germany.
Domestic flights are generally used for business, which tends to make trains a little simpler, and often cheaper, to use. But the boom in budget airlines means that increasingly air fares between major cities can compete with trains. It's definitely worth looking around before you book your travel. Something to be aware of is that low-cost airlines sometimes use small airports a long way from your intended destination, which can end up costing you a lot more in additional travel.
Germany offers a fast, reliable, and largely affordable railway network that reaches most corners of the country. You'll almost always save money if you book in advance, so it's worth planning ahead. Travelling from Munich in the south to Hamburg in the north takes around 6 hours, as opposed to 8 by car, depending on traffic.
Most trains are operated by Deutsche Bahn, and their website is available in a range of languages. It's a great resource for working out transportation inside Germany, as well as further into Europe if you need.
Long distance bus lines in Germany can offer a cheaper way to cross the country, especially if you're going to or from Berlin. There is also a cheap service that, if booked in advance, offers trips connecting Hamburg, Hanover, Kassel, Frankfurt, Mannheim, and Heidelberg for a just a handful of euro. These buses run throughout the night.
Most cities feature extensive local bus services which are great for sightseeing, though its wise to look up your journey before setting off to prevent ending up in the wrong place.
If you have your own car, Germany's road network is world-famous for how well it is maintained, and for the Autobahn, which offers high speed travel without a toll (unless you're driving a truck). Be warned: petrol prices are high due to taxation.
If you want to take advantage of the roads but don't have your own wheels, all German airports offer car hire services, as well as in most cities. You can also hire one-way rentals (as long as you're staying in Germany). Most car rental services won't let you drive their vehicles into eastern European countries, including Poland and the Czech Republic. If those are on your itinerary it could be worth heading there first and renting there - the restrictions don't apply the other way around.
Most hire cars in Germany have a manual gearbox (stick-shift), so if you need an automatic make sure to ask for it.
Carpool services in Germany are popular with some travellers, whereby you arrange connections online and essentially arrange lifts - this is a lot easier if you speak German! If you have your own car you can get involved and take passengers as a way of saving or making some money.