On a generic space, say with line element $ds^2 = ydx^2 + xdy^2$, we can define the invariant inner product $$V^2 = V^\mu V^\nu g_{\mu\nu}$$
Lets say that I now consider a constant vector $V^\mu = \pmatrix{1\\ 0}$.
Despite being constant, it looks as if the covector must now depend on the coordinates: $V_\mu = g_{\mu\nu}V^\nu = (y,0)$. This would imply that the inner product depends on the coordinates. If I now change my coordinates it looks like this will change also, but I don't think it should: I'm obviously confused about something.
Your questin is how vector fields are to be described in arbitrary curvilinear coordinates. To this end we recall that, relative to a rectangular coordinate system, the length $|X|$ of any vector $X$ is given by \begin{equation} |X|^2 = \sum_{j=1}^n \bar{X^j} \bar{X^j} \end{equation} $\bar{X^j}$ are the components of the field relative to the rectangular system. This can be written in terms of our curvilinear coordinate system as follows by introducing the metric tensor $g_{hk}$, so that \begin{equation} |X|^2 = \sum_{h=1}^n \sum_{k=1}^n g_{hk} X^h X^k \end{equation} ${X^h}$ are the components of the same field relative to a curvilinear system. Vectors are the same in any coordinate system. However vector components may change from one coordinate system to another.