What is Buddhism's view about debating teachings?

Should we accept teachings without question?

No never.

Accepting teachings blindly is not part of the Buddhas teaching.

The Buddha stressed over and over, throughout his 45 years of teaching, than we ourselves must work. No one can do the work for us. The Buddhas can only show the way. We ourselves must walk through the doors. That also goes for the cultivation of insights.

The Buddha taught that one should go and test the teachings for oneself. One should not even trust ones own ideas and thoughts, because they themselves can be decieving. They are subject to the 3 marks of existence, meaning they are impermanent and in a constant flux. One might believe something one day and next week, the opposite mental formations arise.

The way to do this, is to practice insight meditation (vipassana) and gain insight about how reality functions.

This experiental knowledge, serves as a reference point to reality, meaning that one now has a measurement-tool. When one sees for oneself how reality functions, how the mind operates and how the mind-body-complex functions, then there is really no doubt anymore.

When reading something in a book, when being taught something by parents, teachers, friends, authorities, one is only learning this intellectually, meaning that one does not truly and ultimately understand, how reality functions.

One does not have wisdom (paññā), thereby making it blind belief/faith.

In Buddhism we have faith (saddhā) but this faith is guided by wisdom, so that it does not become blind faith. Wisdom, one gains from the practice of insight meditation. One thereby sees with ones own eyes, what this samsaric existence is and how it operates.

If we should debate, what is the correct way of doing it?

I think this question has a variety of answers. I would like to put forth the factor that when debating there should be a reference point. Something that ties the strings together. That denominator is experiental knowledge of reality.

Debating could be done on the basis of experiental knowledge. Debating on the basis of non-experiental knowledge (non-wisdom), would not make it relevant to the winning of Nibbana.

That kind of debate would be intellectual and more of a scholarly approach.

This approach I mention here, is founded in the practical/insight-meditational aspect of Buddhism. Bare in mind, that this is just one approach to debating. There are many others and I don't think one way is more correct than the other. Unless we are talking about winning Nibbana - then only experiental knowledge can set one free.

The Buddha makes it clear that (1) we should always question the teachings, but (2) not in a showy or superficial way. He contrasts the artificiality of the kind of debating skills that are taught in schools with the correct pursuit of wisdom, which involves a deep penetration and comparison of the teachings with the goal of cultivating real understanding and not merely impressing people.

The correct way of discussing spiritual teachings (the Buddha says) is to find common ground and proceed from there on the basis of truth. When we look at how the Buddha actually engaged the Brahmans, for example, he always sought to understand the inner meaning or significance, often ethical, of the practices and the teachings, rather than taking them literally or understanding them superficially. With respect to the dharma, new teachings should be criticized in the context of the established tradition - which itself should be the result of criticism - in order to arrive at right view.

Again, this does not imply a superficial or fundamentalist approach. The Buddha says that teachings should not be accepted simply because they are traditional or taught by one's teacher.

Another way to put your question (re-phrase) is... "Is criticizing other religions allowed in Buddhism?"

If that criticism is fair and aims to correct others, then it’s allowed and is a good thing to do. As it is mentioned above in Cula-dukkhakkhandha Sutta, Jainism is criticized by Supreme Buddha.

Supreme Buddha was one of the first crusaders against caste system, an apartheid evil ideology started by Aryans who used it to suppress indigenous Indian people. Lord Buddha was a social reformer and thoroughly criticized the animal sacrifices and discrimination of women which was common in the Hindu society.

A heartfelt touching discourse of Lord Buddha on animal sacrifice comes in the Sutta Nipata, Brahmana-dhammika Sutta. In this discourse, it is discussed about the ethical conduct fit for a Brahmin. Lord Buddha speaks respectfully of ancient Brahmins who rejected disdainfully the taking of life and never allowed their religious rites to be contaminated by the blood of poor animals. But corruption came upon, and they started the wicked practice of animal sacrifices to please gods. When the knife was laid on the neck of cow, the gods themselves cried out in horror, seeing that crime of ingratitude and insensitivity committed on an animal that was such a faithful worker, such a sustain-er of life, to humans.

In the Agganya Sutta, Supreme Buddha focuses on the Brahman idea from Wrig Veda that Brahmans came through the mouth of Maha Brahma/God; thus they are supreme. Lord Buddha questions the Brahmins as to how they can claim so when they see day to day that women in the Brahmin caste get pregnant, give birth, and take care of their children? Don’t they understand that they all came from the mother’s womb, not from the Brahma’s mouth?

‘One is not low because of birth nor does birth make one holy. Deeds alone make one low, deeds alone make one holy.’ – Supreme Buddha (Dhammapadha)