Question about vector and tensor field

Question

I am trying to understand vector and tensor fields from "Shun-Ichi Amari" but I think I am missing some concepts, let me first write everything that I have understood
$\textbf{Vector Field:}$Let $X:p\mapsto X_{p}$ be a mapping that maps each point in a manifold $S$ to a tangent vector $X_{p}\in T_{p}(S)$ and we call such a mapping a vector field. So this definition I understood.
After this author defined a family $[T_{p}]_{r}^{q}$ for $q=0,1$ when $q=0$ we have
$[T_{p}]_{r}^{0}=\{T:T_{p}\times..\text{(r times)}...\times T_{p}\to \mathbb{R} \text{ such that T is multilinear}\}$ and when $q=1$ we have $[T_{p}]_{r}^{1}=\{T:T_{p}\times.....\times T_{p}\to T_{p} \text{ such that T is multilinear}\}$
then we have a mapping $A:p\mapsto A_p$ which maps each point $p$ in $S$ to some element $A_{p}$ of $[T_p]_{r}^{q}$ where $q=0,1$ a tensor field of type $(q,r)$ so basically the map $A:S\to [T_p]_{r}^{q} ?$ am I right? so if I choose a point $p\in S$ then $A_p\in[T_p]_{r}^{q}$ that is for $q=0$, $A_p:T_{p}\times ...\times T_{p}\to \mathbb{R}$? I think this is what it means, correct me if I am wrong here.
Further we have $A$ be a tensor field of type $(q,r)$ and $X_1,X_2,....,X_r$ be $r$ vector fields.Then we may consider the mapping with domain $S$ of the following form:
$A(X_1,X_2,.....,X_r):p\mapsto A_p((X_1)_p,(X_2)_p,......(X_r)p)$ I am having problem in understanding what this maps is how the element will look like? can someone please explain that how this map is defined.