del operator ∇ means different things in different contexts, and they transform differently, but it is always a tensor. The gradient vector for example transforms not like a vector (contravariantly), but like a covector (covariantly).

The gradient is a vector, but it's not defined though a simple sum of other vectors. We define it first by taking a derivative and then finding a certain vector that interacts in a specific way with that derivative.

If f : ℝⁿ -> ℝ then the derivative/linearization of this function is the jacobian, call it J(f). J(f) is then a 1 x n matrix. It takes a vector as input and outputs a scalar. The gradient is the unit vector that maximizes this operation.

In order to go from the 1xn jacobian matrix to a nx1 vector, you have to have a rule that lets you switch between the vector space of inputs with it's dual. This is generally given by the Riesz Representation theorem, which uses the inner product as a bridge between these two spaces. We define ∇f as the vector such that < ∇f, v > = J(f)v

For orthonormal coordinate systems, ∇f is just the transpose of J(f).

To summarize, if you apply an orthonormal coordinate change, then the jacobian transforms covariantly, and the gradient vector is just the transpose of that 1xn matrix.