My mum shopped at Tesco's when I was a kid and sometimes Somerfield (when it was "Gateway" - anyone remember that?) And in those days you could also go to markets to the meat van and buy really good quality meat cuts and she would fill up the freezer for roasts.
As I grew up she switched to Sainsbury's (used to be the more upmarket store) Tesco's has gone downhill in the last 10 yrs and Sainsbury's has a lot of problems here with their suppliers, so week after week the items you want aren't there. Our Sainsburys is also two bus rides away and then a good walk home from the bus stop with all the bags (taxi would add £8 to the cost) so we gave up on that.
Our nearest store is a Waitrose store (used to be Safeway- HATED Safeway, lots of items that would go off very quickly and bakery useless). So now, we do our main shop at Waitrose (only 2 of us, so I don't think this would be viable for a larger family) but there are enough deals on what we buy to keep the costs down and after months of shopping there, the range does become limited- but then they also carry brands and items no other stores do and it's only a £3 taxi ride home, so that's easy to budget in.
M&S is my favourite (no surprises there) and I don't find a lot of difference between them and Waitrose, so we end up with M&S favourite items (roasted chicken breasts/fresh coleslaw and their fish) as a top , but then we get a nice change when the Farmer's Market is on (once a month) and we can buy local organic produce, although it's not always cheaper than Waitrose, but their strawberries are fab and I have recently got into a smoked goats cheese (over pine nuts) that I love and their fresh asparagus bundles. I should also mention there is a fudge maker who makes the most awesome coffee or coconut fudge (very expensive tho!)
Only been to Asda once, about 5 years ago, the nearest one is about 15 miles away and I wasn't impressed at the time, although I have heard they are very very good now.
Then we get the French Market (bargains on the bread and the pastries, oh lord, you can get about 4 pastries from them, typical fancy french items for half the price of waitrose) and beautiful hand milled soaps etc. One of my faves is the Italian market, we often buy a lot of tomatoes from them, tomatoes like I had never tasted before, not as cheap, but worth every penny and here is where I learned my lesson that quality and flavour come at a price, but you don't require much of it and can have a very simple meal (say fresh bread, tomatoes, cheese and some olive oil) and it can be so delicious- I wish we could shop more like the French in this way- buying a few items every few days from markets and small local shops.
We don't eat a lot of meat, but it's important to me that any meat I do eat is not factory farmed in anyway, preferably organic and I would rather pay the price and have it less often.
In the U.S we had a Stop & Shop which I didn't like, but there was store called "Big Y" that carried rolls that tasted like british ones, a selection of cheeses and various items from England ( to my eternal shame, I once spent $3 on a week old "Sun" newspaper!!) and when I go back I will miss my Waitrose and french markets, but intend to find farmers markets and go to TRADER JOES! I also hope to look into veg box schemes as my friends tell me the organic selections are nowwhere near as good as here yet - and I didn't know Walmart sold meats and stuff, so that is new to me)
Edited to add: Good lord, I went on and on and on, I am sure you're all delighted with my life and times in supermarkets 
