For everyday shopping, most countries offer the same layered ecosystem once you learn to read it. National supermarket chains handle the weekly shop, discount chains save serious money on staples, convenience stores cover emergencies at a premium, and traditional fresh markets often beat everyone on fruit, vegetables, and local produce.
When you first arrive, ask locals or expat groups which chains are considered good value versus premium, because every country has its expensive-but-fancy chain and its cheap-and-cheerful one, and the difference on an identical basket can be substantial. Pharmacies versus supermarkets is another thing to learn: in some countries even basic painkillers or vitamins are only sold in pharmacies, and toiletries may be cheaper in dedicated drugstore chains.
Loyalty cards are usually free and worth getting immediately, as discounts are often locked behind them. Fresh markets typically peak in the morning for selection and late afternoon for bargains. And if you cook food from home, hunt down the ethnic grocery stores early; Indian, Asian, and international shops exist in most mid-sized cities and will save you both money and homesickness.