I went supermarket free for a year. I can tell you it's worth it.
Supermarkets do untold damage to the economy, closing down small shops, I could go on...
I don't know where you live but I live in a built up are where there are scores of local shops and a market nearby
This is what I've found;
Supermarkets put a big margin on incidentals such as light bulbs and things you don't often buy or question the price. By shopping at independent hardware and electrical stores you save a lot.
Asian supermarkets are great for spices ( a fraction of the price) rice, tinned tomatoes,oil, pulses, loo rolls and kitchen towels again much cheaper.
The local market is great for fresh fruit and veg, much of it locally grown and yes cheaper. Meat and fish too.
You will not always find the same variety of convenience foods in local shops but this isn't always a bad thing as you'll need to cook more from scratch..
As for delis I tend to use the polish ones as we have a number of these where I live; great for gorgeous bread £1 for a wheat and rye loaf that makes gorge toast and lasts for ages.
Learn to be creative and think outside the box. You wont be able to do a big shop like you can at Asda, but you'll find it a much more enjoyable experience.
Good luck x