Brand recommendations honestly vary so much by country that naming any would mislead; the more useful skill is knowing how to buy safely anywhere. Stick to sealed bottles from proper shops, supermarkets and pharmacies rather than loose vendors, and check that the cap seal is intact and the bottle looks factory-crisp, since refilled bottles are a known scam in a few tourist-heavy places. Locally bottled water from major national brands is typically fine and much cheaper than imported labels; watch locals in any supermarket and buy what they buy.
For longer stays, endless bottled water gets expensive and wasteful, which is why most settled expats switch to alternatives: large refill jugs delivered to the door, which are standard in many countries, or a home filtration unit. For travelers, purification options have become genuinely good, filter bottles, UV purifier pens and purification tablets all work well, with tablets being the lightest backup for trekking and remote areas where bottled supply is unreliable.
One extra habit: in restaurants in bottled-water countries, ask for the bottle to be opened at your table. It is a normal request that no decent establishment minds.