There was an article recently about a young woman of 17 who had to take over the housekeeping while her mum was being treated for cancer and got the shopping bill for quite a big family down to under £20 a week 
I couldn't work out how she'd managed it but then I realised she was 17 and had no preconceptions about where things should come from so was using muscle food for meat and the market for veg, Asian supermarkets for rice/noodles/spices. None of that is rocket science but most people go to one supermarket and get everything from there regardless of cost.
You could do the same. Work out what your staple starch is, where its cheapest & can you source it locally. Then meat, grocery items, cleaning supplies etc.
If you look on moneysaving expert they have loads of ideas. The biggest savings to be had are often on grocery items and cleaning supplies. Aldi cleaning products are all leaping bunny approved and good housekeeping institute approved. Most are under a pound compared to nearly £3 for branded stuff.
If you set up an account at mysupermarket they have a price watch facility, if your favourite brand is reduced they send you an email. I use it for dhs preferred mayo. Brands I can get away with swapping without him noticing like squash I decant & he is yet to notice.
Don't forget stores like poundstretcher, home bargains and b&m do grocery items too. I stock up on dolmio pasta sauces, always £1, and tilda rice. £1 a bag, its nearly £4 in the supermarkets.