Basis vector & basic one-form coordinate changes in general relativity As I understand it, coordinate axes are defined by the tangents to coordinate curves. The coordinate curves are formed by the intersection of coordinate surfaces.  The nature of the particular coordinate surfaces will determine if the coordinate axis change direction from point to point.

In a book on general relativity, the author has two statements where basis vectors and basis one-forms are introduced, viz., "basis vectors are tangent to the coordinate curves (along which only one of the coordinates changes), and basis one-forms are gradients of the coordinate surfaces (on which only one of the coordinates remains constant)."

It is the parenthetical statements in bold about which I am puzzled.  Specifically, unless all the coordinate surfaces meet orthogonally everywhere, the meeting of which forms the coordinate curves, a restriction not specified in the book, then any movement, even infinitesimal, made along the tangent coordinate axis could change one or more of the other coordinates as well.  That is, aren't the bold statements either assuming orthogonality or are overly general?
The textbook (Collier, A Most Incomprehensible Thing, 2012, $\S5.1$) seems roughly alright, but your intuition is valid too:


  any movement... made along the tangent coordinate axis could change one or more of the other coordinates as well.


Now we can move along a coordinate curve without changing the other coordinates, as seen from your description of intersection of coordinate surfaces. I will use Schwarzschild[-Droste] coordinates $(t,r,\theta,\phi)$ as an example. You could take the coordinate surfaces $\theta=\pi/2$, $\phi=\pi/4$, and $t=0$ say, and then change $r$. This will not change the other coordinates!

But counter-intuitively, the $r$-coordinate curve does not depend on $r$ alone. Contrast Gullstrand-Painleve or Eddington-Finkelstein coordinates, which use identical $r$, $\theta$, $\phi$, but a different zero-th coordinate. Using one of these, the $r$-coordinate curves are actually different curves through spacetime! I learned this by accident, the hard way. (If you take the vector given by components $(0,1,0,0)$ in a given one of these coordinate systems, and transform it to another, you will not get components $(0,1,0,0)$ in the new coordinate system!)

On the other hand $dr$, which as the text says is a basis one-form, is independent of the other coordinates chosen. However I would call it the "gradient of $r$", to use the modern meaning of gradient as a one-form / dual vector. The textbook's language "gradient of the coordinate surface" makes me think of the traditional vector (not dual vector) definition of the gradient, and may mean the tangent $\partial_r$ to the coordinate curve, which should be used with caution as explained above.
The textbook (Collier, A Most Incomprehensible Thing, 2012, $\S5.1$) seems roughly alright, but your intuition is valid too:


  any movement... made along the tangent coordinate axis could change one or more of the other coordinates as well.


Now we can move along a coordinate curve without changing the other coordinates, as seen from your description of intersection of coordinate surfaces. I will use Schwarzschild[-Droste] coordinates $(t,r,\theta,\phi)$ as an example. You could take the coordinate surfaces $\theta=\pi/2$, $\phi=\pi/4$, and $t=0$ say, and then change $r$. This will not change the other coordinates!

But counter-intuitively, the $r$-coordinate curve does not depend on $r$ alone. Contrast Gullstrand-Painleve or Eddington-Finkelstein coordinates, which use identical $r$, $\theta$, $\phi$, but a different zero-th coordinate. Using one of these, the $r$-coordinate curves are actually different curves through spacetime! I learned this by accident, the hard way. (If you take the vector given by components $(0,1,0,0)$ in a given one of these coordinate systems, and transform it to another, you will not get components $(0,1,0,0)$ in the new coordinate system!)

On the other hand $dr$, which as the text says is a basis one-form, is independent of the other coordinates chosen. However I would call it the "gradient of $r$", to use the modern meaning of gradient as a one-form / dual vector. The textbook's language "gradient of the coordinate surface" makes me think of the traditional vector (not dual vector) definition of the gradient, and may mean the tangent $\partial_r$ to the coordinate curve, which should be used with caution as explained above.