(I'm not a native speaker. As follows from others' reactions, native speakers on this site clearly prefer at.)
I am learning the ropes of my new job.
..this usage seemed strange to be. I imagined some real-life ropes. But I've checked Google Books, and found several instances of the phrase being used, apparently by native English speakers:
One phase is the candidate's professional transition (learning the ropes of a new job) and the other phase is the candidate's personal transition (adjusting to life after college in a new ... (Craig Ross, Brent Beggs, Sarah Young - 2011)
I am learning the ropes on my new job.
Google Books attested 3 results for me. Here's one:
Learning the ropes on a new job means getting used to a new routine and meeting new people. (Laurie Nadel, Judy Haims, Robert Stempson - 1992)
I must point out that being attested with a meagre half-dosen results at Google Books would be highly untypical for a typical expression, so these examples might be marginal. On the other hand, I might've used some unlucky combination of search words.
Personally I would use the expression either without any prepositional phrase attached to it, like here:
You'd better find someone to show you the ropes if you're going to fix the car yourself.
.. or with a prepositional phrase that is more attached to "learn" than to "ropes":
It can take quite a while for a new lawyer to learn the ropes in a big legal firm.
or with at:
I am learning the ropes at my new job. (this way, the phrase is also clearly not attached to "ropes": at denotes location, so no ambiguity arises)
But here's one interesting Ngram:
Clearly of is widely used, but it must be used in ways that do not result in ambiguity. I guess the tendency is to use it with prepositional phrases denoting something abstract, some thing that cannot have literal ropes - here are Google Books examples:
And your husband or partner will be exploring his new role as a dad and learning the ropes of child care. ("Child care" cannot have real-life ropes; it's an abstract concept)
Zimin is learning the ropes of being a house church pastor. ("Being" is also a concept that cannot have real, hemp-and-tar ropes.)