How is a basis induced on a vector space?

Question

I'm studying the concepts of the tangent and cotangent space, and I'm wondering how bases are induced, and what that means concretely. I feel like I have an intuitive guess as to what it means, but how do coordinates in the tangent space induce a basis set there? How is that rigorously established? Is there a general way that components allow you to deduce what a basis set on any vector space will look like? Further, how does a basis in the tangent space induce a basis on the cotangent space? Again, I feel like I intuitively understand why the basis is the differential, but I don't know how to derive it for things to click in my head.