And why not go to France? In Saint-Louis, just across the border, you've got some nice supermarkets: Géant Casino, Leclerc. Only difference with Germany is to get the tax exemptions (starting at 175 euros in France). And of course the language, as you can perhaps be better off with english in Germany...
closest to go serious shopping is Lörrach, which can be almost reached by Tram 6. Tram 6 (direction Riehen) gets you to the border. From there you can walk over the border, and either walk to Lörrach (about 2km) or take the bus. It will take you about 45mins in total.
(note that you can go to e.g. Grenzach for grocery shopping, and Weil-Am-Rhein as well, but those are much smaller places)
You can buy bus tickets on the self-service machine on the bus stop.
When buying clothes, groceries, etc, you can claim VAT back. When paying ask for it, and they will provide you with a filled-in form that you only need to stamp at the border. (do not forget your CH residence permit, otherwise you will not be able to do it). Next time you visit the same shop, bring the form and they will give you the cash.
The only advantage of France for me is selection. It's no cheaper. Germany on the other hand is cheaper and it's good. If you go shopping in St. Louis, do stop at Boulangerie Paul just up the road. :-)
France
St. Louis - Geant Casino
Germany
Lörrach - Hieber
Here is a site that lists various shops in Lörrach, Germany.
There are shopping malls all around; in Germany, in France and in Switzerland.
I wouldnt worry too much, if I were you. The opportunities for shopping here are abundant. It will take you some time to find things that you like and dislike, to figure out a shopping budget, and to do the VAT claim back for your shopping in Germany and France. It is all easy, and someone of the friendly EFers will show you how it is done once you are here.
Driving into Germany? Go along the Hiltalingerstrasse in Basel, drive across the border and 50 meters on the left is an enormous shopping center with free multi storey car parking, about 10 storeys! Shops are on the first 2 floors. Building is at junction of Zollstraße and Hauptstraße in Weil am Rhein.
Going to Lörrach, I would take the S Bahn train from platform 1 or 2 at Basel SBB station. There is a German ticket machine near platform 2 I think it costs €2,70 for a single ticket (the SBB will also sell you a ticket) Take the train to Lörrach Hauptbahnhof, about 4 stops.
There is a good Indian restaurant "The Ganges" very near the station. If you get homesick, you can also find a Migros in Lörrach!