Previously I didn't understand why $\frac{\partial }{\partial x^i}$ can be a basis for a vector field $X$. For example, in integration of curve length, if we have a curve $\in \mathbb{R}^2$,$\lambda=(x^1(t), x^2(t))$ or $(t,y(t))$, then $\frac{dx^1(t)}{dt}, \frac{dx^2(t)}{dt}$ or $(1,y'(t))$ is vector along the curve and so (part of) the vector field. It seems the basis should instead be something like $\frac{dx^i}{dt}$ or just $dx^i$ (since the denominator is a real number and don't change direction of the basis vector).
Then I find an explanation for $\frac{\partial }{\partial x^I}$: In differential equation, taking a case of $\mathbb{R}^2$ as an example, we have vector field of an integral curve, so we may say vector field $X$ is a function of an integral curve $\lambda$, $X(\lambda)$. And it equals differential of the curve, say $\frac{d \lambda}{dt}$. So we have $X=\frac{d }{dt}$. It seems therefore the basis for $X$ should be $\frac{\partial }{\partial x^i}$.
The two kinds of basis both seem to make sense, but obviously they are completely different.
Though I know that $dx^i$ is the basis for dual/covariant vector, and I may well accept that conclusion, I still can't see what actually goes wrong in my thinking. So what happens?
(Edited to add:)
Here is a note for myself: I see this question seems to be more complicated than it appears and is extensively discussed in the chapter about tangent bundle in Spivak's book. It's about equivalence between different tangent bundles (candidates) and there is, furthermore, a theorem to prove such equivalence. I see relations between these candidates are the thing I'm currently considering, and relation between tangent bundle to cotangent bundle is another issue (and should not be mixed up with this question).