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The ways through which knowledge is acquired are three: perception, the report, and reason. We know those ways of knowledge are reliable, because when we recall the information that we know, we find that all of it has been acquired through one of those three channels.
Perception
Perception involves the acquirement of information through the senses. The senses are divided into the inward, and the outward senses.
The inward sense is introspection, by way of which you are able to acquire knowledge about yourself. For example: when you feel hungry, you know that you are hungry by introspection.
The outward senses are five: sight, hearing, smell, taste, and touch.
When the perceived object is detected with one of the five outward senses, a realization emerges in the body which the mind then interprets. The reality of perception, is this emergent realization.
For example: when light reflects off the surface of a body and enters the eye, the eye reacts and a realization emerges within it. This emergent reaction is what is called “sight”.
Sensory realizations correlate with what is perceived, since they are reactions to it. Through the mind’s interpreting of this realization, knowledge is acquired.
Notice that the outward senses, in of themselves, do not make mistakes. They are merely a window connecting the mind to the outside world. Rather, mistakes occur when one misinterprets sensory realizations.
Reports
Reports are accounts of events that we did not personally perceive, communicated to us by those who did perceive those events.
For a report to be a way to knowledge, it must be mass transmitted. Mass transmission (Tawatur) is a report’s being transmitted to us by so many independent witnesses, such that it would be nomically impossible[1] for them to have all conspired to fabricate it.
For example: consider the existence of a foreign country that you have never been to. Despite not having personally visited this country, you still know that it exists. This is because you have received reports about this country’s existence from various sources: whether they be reports from people who lived there, or people who visited it, or even pictures of this country that you’ve seen on TV. All of those reports, when taken collectively, form mass transmitted proof for the existence of said country.
Reason
Reason is the human ability to use non-inferential knowledge, in order to acquire inferential knowledge.
For example: once the meanings of “body”, “rest”, and “motion”, are established non-inferentially for the thinker, he can come to know that “a body cannot exist in a state of being neither at rest nor moving”.
[1] “Nomically Impossible” meaning: a proposition whose truth violates normalcy.
More on nomic judgements here.
Keys to the Unseen 
How do we know that sense perception is reliable?
I can get that reason is reliable in virtue of logic (rules of inference and the laws of logic) but what proves sense perception to be reliable?
It is established that not everything known was inferred. If it were, everything known would require a proof, and knowledge of this proof would require a proof for the proof, and then a proof for the proof of the proof… and so on and so forth ad infinitum.
Thus, there are things which are known to be true without requiring a proof to establish them to be so. See more on this here. And our position is that the three ways of knowledge described above are known to be ways of knowledge by every Mukalaf in this manner. As such, no Mukalaf requires a proof for their reliability.
As for the one who claims to not accept any of the aforementioned ways of knowledge, then we say: either this person is sincere in his claims (i.e. he claims to not accept them as ways of knowledge, and he genuinely does not believe that they are ways of knowledge), or he is insincere in his claims (i.e. he claims to not accept them as ways of knowledge, when in fact he believes that they are ways of knowledge).
To the former we say: you are not a Mukalaf. So there is no need to discuss anything with you. This was discussed here.
To the latter, then the classical scholars prescribe that his wealth be confiscated from him, and that he be beaten and made to feel pain. So if he asks for his wealth back, he is told: “why do you ask for something which you believe does not exist?” and if he complains from the pain, he is told: “your complaints are merely reports about your experiences, and reports are not a way of knowledge according to you”. See Abd Al-Qahir Al-Baghdadi’s response to some of the sophists here.
Obviously what the classical scholars prescribe cannot be done today. And unfortunately we are living in times where such silly sophistry is praised as intelligence and critical thinking. So today we would say to someone belonging to the latter group of insincere individuals: “if you’re an insincere person who wishes to avoid discussion by lying about what he believes in, then there is little point for any discussion to happen.”
And so either way (i.e. whether the objector is sincere or not) a discussion on theological issues would be a futile exercise, so the one who claims to doubt any of the three ways of knowledge described above is not worth engaging with.