Jayhawker Soule, as these are matters that deal with the science of epistemology(pramana shastra or Nyaya shastra) there are fewer people who are learned here in in this area. Most Hindu people, have a simple everyday understanding of Hinduism, doing your daily puja, going to temple and duties etc. Few have made an attempt to read the shastras. In fact, though here you may find most of Hindus have read the Bhagvad Gita, a majority of Hindus have not. There are a few people on this forum who are educated in pramana shastra and I am one of them, although I wouldn't call myself an expert on Nyaya either. So anyway, to answer your question.
The three main pramanas or means of knowledge that most schools of Hindu philosophy accept are 1) Perception 2) Inference and 3) Testimony. Others add a few more, but usually these are special types of inference. Such as 4) Analogy 5) Presumption and 6) Non-apprehension. However, the beauty of this epistemological system is it is based on perception. This is a surprise for many who first encounter Hindu philosophy, they realise Hinduism is grounded in empiricism, in much the same way science is. This is how it works:
1) Perception. Defined as union between the knower, the instrument of knowledge and the object of knowledge e.g. I am the knower, my instrument of knowledge is my eyes and the object of knowledge is the tree. There are three types of perceptual knowledge:
1.1. Sensory perception, from the 5 senses or extensions to the senses like telescope
1.2. Mental perception, from the 6th sense or mind, such as perceiving ideas, feelings, thoughts etc
1.3 ESP, from subtle senses, such as clairvoyance etc
2) Inference. It is after you perceive relationship between two or more things. It requires three things; a major term, a middle term and minor term e.g. In the inference there is fire on the hill because there is smoke, the fire is the major term, the smoke the middle term and the hill the minor term. There are three types of inference
2.1. From seen cause to seen effect e.g. It is going to rain, because I see the rain cloud
2.2 From seen effect to seen cause e.g. It must have rained, because I see rainbow
2.3 From seen effect to unseen cause also known as inference of common observation e.g. There must be a force between the magnet and the iron filings that attracts or pushes them, because it is a common observation that nothing moves or changes direction without a force
3) Testimony. Testimony are inferences of word meanings where words denote things which have been experienced. e.g. "There is a huge tower called the Eiffel tower in Paris; I had a dream of such and such tonight; I see auras" It is of two kinds
3.1. Ordinary testimony of laymans
3.2 Expert testimony of trustworthy people
The rest are types of inferences, but while the earlier ones are inductive, these are deductive:
4) Analogy. It is when you recognise something in your own experience based on testimony e.g. I describe to you an animal called a "tiger" which is dangerous and you must get away from it. It is yellow and black striped, it has whiskers, its has hazel eyes, it is has four legs. Then you see such a creature fitting this description and you say "I have seen a tiger" I describe to you a state of experience called an OOBE where you float out of your body, can see 360 degree vision and colours will be very vivid. Then you one day have an experience that fits this description and you say "I had an OOBE"
5) Presumption. It is when you need to posit something to make sense of something else e.g. A person who needs glasses to see is walking around today without glasses. How can he see without glasses? He must be wearing clear contact lenses.
6) Non-apprehension. It is when you know something by negative evidence e.g There are two rooms in a house, room A and room B. John is at home, but is not in room A. Therefore, John is in room B.
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