A nice, classical, way of understanding Tensor Products is as the image of a universal mapping $\phi$. That is, given a family of vector spaces $V_1,\cdots V_k$, the pair $(\phi,V)$ is a tensor product if, for all $\psi\in\text{Hom}_{\mathbb{F}}^{k}(V_1,\cdots,V_k,W)$, there's a unique $\tilde{\psi}\in\text{Hom}_{\mathbb{F}}(V,W)$ such that $\psi=\tilde{\psi}\circ\phi$.
That does make it intuitive to understand what the tensor product is actually about. I mean, you could look at it as something that "linearizes" universal multilinear maps, that is, it serves as a domain for the "linearizations" of all elements of $\text{Hom}_{\mathbb{F}}^{k}(V_1,\cdots,V_k,\mathbb{F})$ simultaniously.
That's nice and all, and gives you a good reason to see tensors as elements of tensor products, and vector spaces as tensor products for that matter.
One thing's still bugging me, and that's the dual space, that is, the space of linear forms of $V$ on $\mathbb{F}$. When defining tensors, you always use the duals, either if it's the tensor as a multilinear mapping: \begin{equation}T:V^{\times q}\times (V^{*})^{\times p}\to\mathbb{F}\end{equation}
Or the tensor as a element of a tensor product: \begin{equation}T^p_q(V)=V^{\otimes p} \otimes (V^*)^{\otimes q}\end{equation}
Which both says the same thing. I'm yet to get why do we have to include the dual in the products.
I know some facts, but I can't connect them together and make proper sense out of the definition. Firstly, every vector space it's canonically isomorphic to it's double dual, and that makes it easy to show that there is a correspondence between tensors and multilinear mappings. Okay. But that shows that it works to put in the dual, not why did we think about it in the first place.
Secondly, I've came to know that a $(p,q)$ tensor is regarded as being contravariant of order $p$ and covariant of order $q$. OK. That makes it nice to see what the dual is doing, because, since it's connected to the covariant vectors, when alongside the contravariant ones, makes us free to not worry about the transformations on the basis, that is, in the sense of not actually having to define a specific basis. But, I don't know why are dual vectors related to covariance.
Lastly, a transpose of a linear map $f:V\to W$ is defined as: \begin{equation}f^{*}:W^{*}\to V^{*}\end{equation} Such that $f(\phi)=\phi\circ f$ for all $\phi\in W^{*}$. Now, every single thing about this definition makes it strongly seem like it's connected to what I'm trying to understand, but I'm yet to make something out of it.
Looks like that's the only thing keeping me from moving on with tensors, so, any help will be appreciated.