If you are concerned about costs (and I assume "plays hell with the budgets, though." means that you are), the single most expensive form of transport in Germany is a rented car (taxi is more expensive on long distances, but on short distances a rented car is more expensive).

As others have already pointed out, there are trains going to most places you would like to visit. The German railroad offers various discount tickets that you can read about here: http://www.bahn.de/i/view/GBR/en/prices/index.shtml (in English).

You can buy various tickets that let you get off the train when you see something interesting, then continue the ride later. The tickets are also valid on many bus lines, so that gives you even more options. You will also find that most workers you will meet in the trains speak English (conductors, ticket sales, information, etc.). On the motorway, you will have a lot of difficulty getting information at road stops and ticket plazas in English.

And finally, if you drive, you will find that many of the road repairs on the national roadways are done during the warm weather (summer). I know this because I used to drive and after sitting still for a long time on a hot day in a rented car, I decided to take the train. I did and have never been tempted to go back.