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Forget the big advertising posters in the newspaper. They're just bait to get you in the door so you end up buying everything you see anyway. It makes way more sense to take your favorite supermarket chain and actually check the prices of the stuff you buy regularly there - milk, bread, meat, whatever. Not what's on sale right now, but your standard purchases. Then you compare that every couple of weeks across two or three other stores. After a while you'll see for yourself where it's worth shopping.
There's also a huge regional difference in this - around here the discount stores weren't competitive with regular supermarkets for a long time, but that's changed a lot. Nowadays the quality is often just as good and the prices are simply better. But not for everything. For certain brands or organic stuff you sometimes pay almost the same as you would in the big supermarket.
Honestly, the most important thing is just not to shop emotionally. "Ooh, 50% off smoked salmon" - cool, but do you actually need it right now? That ends up costing you more money than if you go in with a plan and only buy what's on your list. That's how I do it and I'll sometimes drive to the next store if the price for my regular stuff is noticeably better there.
I think the most important thing is to actually sit down and compare what YOU personally buy. A lot of people just stare at the big deals, but those are often on stuff you don't even need. For example, if you buy a lot of organic products or brand names, then Aldi and Lidl aren't always that much cheaper than other stores because they have less selection. Rewe and Edeka make up for it with their weekly flyer deals that are often really good for specific items.
My trick: just write down what you buy regularly (don't go overboard, just your top 20 items). Then you compare at the different stores near you to see how much your typical shopping trip costs. Some stores are cheaper on fruit and veg, others on dairy products. The best deals also change with the season. In summer tomatoes are cheap everywhere, but in winter it might be worth checking out a different store. And don't forget: a lot of stores now have apps with extra deals that you only get with a loyalty card - you can save a decent amount if you're shopping there regularly anyway.
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