I think the ideas behind the coordiate-free formulation of tensor calculus make it relatively easy though.

A tensor is a function that takes an ordered set of N covariant vectors (i.e. row vectors) and M contravariant vectors (i.e. column vectors) and spits out a real number. It has to be linear in each of its arguments.

I'm pretty sure all the complicated transforms follow from that definition (though you may have to assume the Leibniz rule - I can't remember), and from ordinary calculus.