What are some relationships between a matrix and its transpose? All I can think of are that
If symmetric, they're equivalent
If A is orthogonal, then its transpose is equivalent to its inverse.
They have the same rank and determinant.
Is there any relationship between their images/kernels or even eigenvalues?

Fix a ring $R$, and let $A \in M_n(R)$.  The characteristic polynomial for $A$ is $$\chi_A(x)=\det (xI-A),$$
so that $\chi_{A^T}(x) = \det (x I -A^T)= \det ((xI-A)^T)=\det(xI-A)$.  Since the eigenvalues of $A$ and $A^T$ are the roots of their respective characteristic polynomials, $A$ and $A^T$ have the same eigenvalues.  (Moreover, they have the same characteristic polynomial.)
It follows that $A$ and $A^T$ have the same minimal polynomial as well, because the minimal polynomial arises as the radical of the characteristic polynomial.
