I'm new to tensors, and I need to understand what a certain basis actually is, how to visualise it.
Say we have the $r$-dimensional vector space $T_p M$ and $n$-dimensional dual space $T^{*}_p M$. They have bases $\big\{ \frac{\partial}{\partial x^{b_1}},...,\frac{\partial}{\partial x^{b_r}} \big\}$ and $\big\{ \mathrm{d}x^{a_1},...,\mathrm{d}x^{a_n} \big\}$, respectively (right?).
Now a $(r,n)$-tensor is the map $T: \otimes^r T_p M \otimes^n T^{*}_p M \to \mathbb{R}$, which can be expanded in a basis $\big\{ \frac{\partial}{\partial x^{b_1}} \otimes ... \otimes \frac{\partial}{\partial x^{b_r}} \otimes \mathrm{d}x^{a_1} \otimes ... \otimes \mathrm{d}x^{a_n} \big\}$.
How to 'write out' the basis i.e. what are it's elements? How to visualise it?
Hint: Take, for example the mixed $(2,1)$-tensors $$T={T_{i_1i_2}}^{j_1}dx^{i_1}\otimes dx^{i_2}\otimes\frac{\partial}{\partial x^{j_1}},$$ as tri-indexed linear combination on the basis $dx^{i_1}\otimes dx^{i_2}\otimes\frac{\partial}{\partial x^{j_1}}$.
These basis, as tri-linear maps $T_pM\times T_pM\times T_p^*M\to{\mathbb{R}}$, work via $$dx^{i_1}\otimes dx^{i_2}\otimes\frac{\partial}{\partial x^{j_1}}\left(\frac{\partial}{\partial x^{a_1}},\frac{\partial}{\partial x^{a_2}},dx^{b_1}\right)=\delta^{i_1}_{a_1}\delta^{i_2}_{a_2}\delta^{b_1}_{j_1}.$$