It will depend on where you live. For regular shopping something like Maruetsu or Life (at least in the area more central in Minato). Coupled with Seijo-ishii if you want the occasional splurge. That is then supplemented by a regular shop at Hanamasa to do slight bulk shopping especially for meat and fish (although this is nothing like Costco). However, there are tons of different supermarket brands around and will essentially depend on where you live. As an expat i'd also go to Nissin World Delicatessen occasionally (once every 1.5 months or so) to stock up on European imports that are harder to find in your average Japanese supermarket (especially deli meat/cheese and wine).

Combinis (7-11/family mart/lawson) are essentially that - convenience. I would pick up a ready to go meal or even dinner on the day if I don't feel like cooking, but they are not replacements for a grocery store/supermarket. They are also more expensive as you'd expect for the convenience (although again not absurdly so). Moreover you can easily find the smaller convenience version of a larger supermarket, such at Marutesu Petitle, which are akin to Tesco Express or Little Waitrose in the UK, which would also just about trump any Combini in terms of grocery and even prepared meals sometimes. And they can be just as convenient if not as numerous, since their footprints is a lot smaller than your bigger supermarket, and just a little bigger than a larger combini.