First, we have to assume that we exist. This is the heart of what Descartes was getting at. "I think, therefore I am." We cannot prove our existence, but we have to assume it. Because we think we exist, we must assume that we really do exist. The alternative is that we don't exist, and if we don't exist, any thoughts we may have on reality are really quite pointless. If we don't exist, then there is no point to anything that follows.
Second, we must assume that there is a objective reality external to and separate from our thoughts. It is possible that everything we perceive as reality is just a product of our very vivid imaginations, but if that is the case, then any quest to learn about the nature of that reality is really just an elaborate thought exercise. What you perceive to be a mountain, I could perceive to be an elephant, and if we are both truly living in our own little thought created bubble worlds, then we are both right. If that is the case, then any knowledge we gain about the world only applies to us, but it is a fundamental assumption that we are part of a larger objective reality. We reject the idea that gravity could change simply because I imagine it to be different. There is a reality that is universal to all of us. We may differ in our interpretations of that reality, but we have to assume that it does exist.
Third, we have to assume that our senses are generally trustworthy. Everything we know about reality is filtered through our senses. Now, there can be exceptions to this rule, like a blind person's sense of sight, but as a general rule, in the absence of evidence to the contrary, we have to assume that the majority of what we see is real. Any other assumption is ultimately a fruitless exercise, because it causes our shared universal reality to be unknowable. Without reliable senses, we are right back to the problem of you see a a mountain where I see an elephant, but now it is entirely possible that we are both completely wrong, and there would be no reliable way to figure it out.
So these are a few of the key assumptions about our existence and the nature of reality. Hopefully, you can see why these assumptions are necessary foundations to the pursuit of knowledge. Without these things, any knowledge we gain is either useless or pointless. I think it is useful to list these assumptions, because it establishes common ground in any discussion about the nature of reality.