Wait until you get to Germany. If you want to save money, take the car back to the same place you picked it up.

Find one place in the Black Forest, one in the Pfalz and one in the Mosel.

In the Pfalz I recommend Bad Durkheim.

Black Forest is basically forest, go for walks etc. Pfalz is basically a big wine growing area with the best grown up against the mountains. In the Mosel it depends on what you like to do, there are trains and ferries going up and down the river and a bicycle path along it so no real need for a car here.

A week is a nice amount of time. Don't know what you're looking for exactly, but the Black Forest is great if you appreciate the outdoors, great scenery, the occasional spa town, and handsome old world villages.

A car will allow you a wider choice of accommodations. You might have a look at the Gutach area, maybe the heart of the BF and the home of the Vogtsbauernhof - here's a page from Bavaria Ben's site that describes it briefly and provides accommodations links:

http://www.bensbauernhof.com/gutachimschwarzwald.html

That said, I'm a huge fan of seeing the BF by train. I've been several times and have always found good places to stay within walking distance of the rail stations. There are numerous rail lines that criss-cross the area, including the scenic Black Forest Railway:

Sketch of rail lines: http://www.schwarzwald.com/karte/bahnlinien.gif

BF railway description: http://www.black-forest-travel.com/places-of-interest/black-forest-railway.html

(The BF Railway is line #4 on the sketch above.)

Schiltach is a gorgeous town on line #10, a great town to visit or use as a base:

Gengenbach is a very handsome old-world town near Offenburg on line #4, the BF Railway; it makes a great base for daytrips by train to Gutach and the other BF villages, to Strasbourg FR, and to Freiburg.

Gengenbach: http://www.stadt-gengenbach.de/en/tourism/old-town/

Besides the extensive train network, staying in a BF village comes with the added incentive of completely free rail travel throughout the area. Hosts in Gengenbach and other participating villages will issue you a free transit pass called the KONUS card:

http://www.blackforest-tourism.com/konus

The double-decker trains that cover the BF railway are great for sightseeing:

http://www.bensbauernhof.com/russschwarzwaldbahn.jpg

If you want a glimpse of the areas you'll be in, there are lots of interesting trip reports for the BF and the Rhine/Mosel area at Ben's website:

http://www.bensbauernhof.com/tripreports.html

I see my link to Schiltach above is missing. Here it is:

http://www.evaway.fr/s3/evawayus/photos/original/7B/7B-92-5B-EF-06-D3-3F-04-0F-8F-C3-F8-F2-1A-EF-94-44-78-58-CD.jpg