Unfortunately there is little in the way of hard-and-fast rules on which place names take a definite article, and to make matters even worse, common usage often disagrees with the official position of relevant governments.
This article from the BBC provides some examples:
There are many other country names that are habitually referred to with "the", such as Congo, Gambia, Yemen, Lebanon, Sudan, Netherlands, Philippines and Bahamas.
But according to several authoritative sources, such as the CIA World
Factbook, the Times Comprehensive Atlas of the World and the US
Department of State, only two countries, The Bahamas and The Gambia,
should officially be referred to with the article.
The two Congos are officially Democratic Republic of the Congo and Republic of the Congo. And the longer, official name for Netherlands is Kingdom of the Netherlands.
And a rule of thumb on common usage:
In some of the other cases, says Ashworth, it's largely a question of
usage and how people refer to them. Quite commonly, definite articles
are attached to areas where they have a mix between geophysical names
and a physical entity.
"Groups of islands like the Maldives and the Bahamas. You wouldn't say
'I'm going to Maldives, you'd say 'I'm going to the Maldives' because
it's a geographical area."
Countries like the United States of America and the United Kingdom
also carry the definite article because they are compound nouns with
Professor Liberman says the habit of putting "the" in front of place
names is heard throughout the English-speaking world and is common to
Germanic and Romance languages.
"In general, use of the definite article is unpredictable. Why should
it be London but The Thames? There is no logic for it yet this is the
way it is.
Note that whilst this mostly agrees with the advice from grammarly, but handles the case of the United States and the United Kingdom differently.
In the case of the two main islands of New Zealand, the common use of the article aligns with the rule of thumb from the BBC that compound nouns with an adjective often carry the definite article. According to the New Zealand government however, the northern & southern islands should be referred to without an article as North Island & South Island respectively when the English name is used. The official Māori names (Te Ika-a-Māui & Te Waipounamu respectively) do include the Māori definite article "te" though.
So which should you say?
For most purposes, I would use a definite article here, and it is certainly what I have heard the most in the UK (including from New Zealanders). In official contexts, especially if in New Zealand, I might follow the New Zealand government and drop the article.
I will add that the sentence "we go to the North Island" suggests that this is something you do regularly (it is an odd quirk of English that for most verbs* the present tense describes a habitual actions or statements that are true regardless of the time, rather than a single action taking place in the present). If you are instead talking about a specific trip that is not part of a regular habit we would say "we are going to the North island" using the present progressive.
*Exceptions are copula verbs like to be, to appear, & to seem; verbs of sensation like to see (but not its more active counterpart to look), to feel & to think; and modal verbs like can, may, & should (modal verbs don't have a present participle so cannot form the present progressive).