Finding good veterinary care abroad is usually easier than people fear, at least in cities. Capital cities and expat-heavy areas almost always have veterinary clinics with English-speaking staff, and university veterinary teaching hospitals, where they exist, offer high-end care and usually some English. Smaller towns are hit or miss, which is worth factoring into where you settle with a pet.
The reliable ways to find them: expat community groups and pet-owner forums for your city are the fastest route to current recommendations; your embassy's website sometimes maintains lists of English-speaking service providers including vets; and international veterinary chains operate in a number of countries.
On arrival, do a routine registration visit while your pet is healthy: the clinic gets your animal's history, you learn the emergency procedure, and you have a relationship before any crisis. Bring translated vaccination records, and ask about local parasite risks, since diseases vary by region and prevention protocols differ from home.