In Riemannian geometry, the concept of covariant derivative/connection have at least 5 different equivalent definitions, as listed in Spivak's book. Most text books, like do Carmo' or S.S.Chern's et, start from koszul connection, then extend to other concepts.
But the most natural and beautiful way is what Arnold did in "Mathematial methods in classical mechanics". Using concept of geodesic, he defines paralle translation first, then Riemann curvature, then covariant derivative. But since the text is in appendix, he skips too many important proofs of many important conclusions. I'm not strong enough to fill them all.
I'm so eager to find a text book expanding the concept philosophy in Arnold's way, with all the elaborated details. But I cannot find the book. Can someone tell me which book satisfy my taste? Thank you so much.
--update-- @Deane 's comment reminds me that Arnold is not the only one who used geodesic as the start of the whole concept logic. Misner as well, in "Gravitation", where the geodesic is defined in a physical way, "free fall trajectory in curved spacetime". This is not rigorous from mathmatical point of view.