It may be possible to save time and simply skip classical differential geometry -- I studied some of it first from [3] and as part of [4].

As with most modern maths familiarity with differentiable manifolds is essential. This is very different from what is taught in schools and undergraduate courses and probably requires several attempts -- the definitions and examples can be found in [1], further examples in [2].

A basic understanding of principal bundles and associated bundles, which I again got from [1] and, geared towards application, in [5]. Also [6] might be useful.

Principles of Lie groups and algebras, also very quickly from [5], chapter 4.10.

I also briefly looked at [8] regarding bundle geometry.

The key to the modern point of view is the tangent space of the frame bundle (and to understand the meaning of horizontal and vertical tangent vectors). With this basic knowledge it is then useful to study [7], which separately explains and compares all three approaches in differential geometry (Riemann, Cartan, Ehresmann).

References

[1]  S. Sternberg: Lectures on differential geometry