Hi Orla,
I also recommend Herblingermarkt. There you have a pretty good-sized Migros and Coop (although the Coop is now somehow limited because it's undergoing refurbishments. opens again in full size in september).
Those are basically the 2 supermarkets I've used in the last 3 years, they're near my home and the parking's free. Edit: just noticed you don't care about parking, but this is reached by the number 5 bus
I think many people do go to Germany (at least it looks like it from the amount of german supermarket leaflets I get at home, hehe), but I've never done it. There's a limit to what you can bring, which I don't know, and I'm always afraid to be stopped by the police at the border (though that never happened before....).
I don't know any farmer's markets though. There is one in the city centre, I think on thursdays and saturday mornings, it's a building near the parking behind St.Johan's church (the one in the centre), but I've never been inside, just saw the banners on off-days. Maybe give it a try?