On major public holidays, the reliable closures almost everywhere are government offices, banks, post offices, embassies, and most professional services. Schools and many businesses close too, and on the most important national or religious days, even supermarkets and malls can shut entirely, though this varies a lot; some countries keep retail open on all but one or two days a year, while others close tightly every Sunday, let alone holidays.
What usually stays available: emergency services, hospitals for urgent care, convenience stores, petrol stations, and restaurants in tourist areas, though often on reduced hours. Public transport typically runs a Sunday-style reduced timetable. Pharmacies operate rotating on-duty systems in many countries, with the schedule posted on pharmacy doors or online.
Planning is mostly about anticipation. Check the official holiday calendar when you arrive, withdraw cash and shop for groceries a day or two before big holidays, refill prescriptions in advance, and never schedule visa or banking business adjacent to one. Also watch for multi-day festival periods, common in Asia and the Middle East, when the practical shutdown lasts far longer than the official dates suggest.