Generally, it might be a good idea to register the name or logo of your brand in those countries in which you are currently operating or are planning to operate in the future. The bigger the customer base you have in a given country, the higher the chances your brand might get copied.
You can always start with one country and expand your registration in the future. In each country/jurisdiction, you'll have to file a separate application. Once your customer base and your sales grow, you can ask your legal representative to file your trademark in more countries. In case your application fails in one jurisdiction, it does not mean it will fail in the other. Every application is examined individually by an examining attorney of a respective IP office of a given country.
It is crucial to file your trademark application as soon as possible to gain priority right over anyone who might file an application with an identical/confusingly similar name or logo. The important thing to note is that even if the registration process takes the IP office a few months to complete, your brand protection starts from the date of filing once it is successfully registered.