Studying tensors, I get the concept of a contravariant and covariant vector, but my understanding is that a contra/covariant basis is different. Why are they perpendicular to each other or what is the motivation? Shouldn't the idea of the variance of a vector apply the same way to a basis vector?
On a similar note, why is it that the contravariant components are described by adding together the basis to form a vector, but covariant components are found by a dot product? What makes the dot product significant here?
Maybe you are confusing three kind of pairing (dot products) among vectors and covectors.
If you pair a pair of vectors $v\cdot w$ is not the same if you pair a pair of covectors $\langle f,g\rangle$. But in mixed case of pairing a pair, $f$ a covector and $v$ a vector, one can choose $f\bullet v=f(v)$.
But in case of choosing basis $b_k$ for $V$ and $\beta^l$ for $V^*$, is then when we have $$\beta^k(b_l)=\delta^k{}_l,$$ which the important device to access to the concepts and techniques about tensors.
Pairings enjoy the property of been bilinear maps i.e. rank two tensors.