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Cosma Shalizi has documented their own pathway into ergodic theory, with an eye towards applications in statistical learning theory (but with many side branches). You might find the annotated bibliography helpful.

More specifically, I understand the question as being motivated by a plan to embark on original research at the interface of ergodic theory and stochastic analysis. For such a purpose it seems helpful if the topic is less mature, less well established, and perhaps less well covered by "landmark papers".
One example: Ergodicity economics is a branch of mathematical finance at this interface. A foundational paper is by Ole Peters (2019). There are intriguing connections with machine learning.