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How does one distinguish between the transformation properties of a scalar field $\phi(\textbf{r})$ or vector field $\textbf{A}(\textbf{r})$ (more generally, the tensor fields) from the transformation of ordinary scalars or vectors (which are not fields in the sense that they are not defined as the function of space)?

EDIT: For simplicity, I want to understand the transformation rule of constant vector and and vector field under rotation. Under rotations, the components of a constant vector $\textbf{A}$ transforms as: $$A^\prime_i=R_{ij}A_j$$ Does the same rule apply for vector fields too? Will it not matter that the argument of $\textbf{A}(\textbf{r})$ also transforms under rotation?

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Yes it does affect the argument as well. The full transformation rule is given by: $$ \mathbf{A}(\mathbf{x}) \rightarrow R\mathbf{A}(R^{-1}\mathbf{x}) , $$ where $\mathbf{A}(\mathbf{x})$ is the vector field as a fucntion of the position vector $\mathbf{x}$ and $R$ is the rotation matrix.

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