Bringing a pet across borders is very doable with planning, but the lead time is the part people underestimate; for some destinations preparations start months in advance.
Where it gets stricter: rabies-free islands and countries typically add a rabies antibody blood test with a mandatory waiting period after it, import permits, and in the strictest cases quarantine on arrival, ranging from days to months if paperwork is imperfect. Some countries ban certain dog breeds entirely, and birds, rabbits and exotics follow completely different rules from cats and dogs.
Your two authoritative sources are the destination country's agriculture or veterinary authority website and your own country's export requirements. Follow the checklist in exact order, because doing the vaccine before the microchip, for example, invalidates everything. A vet experienced in travel certificates, or a pet relocation agent for complex routes, saves enormous stress.