The phrase 'learn your way around' is correct and usable in written English.
You can use it when you are trying to advise someone to become familiar with a particular environment or situation. For example, you could say, "If you're new to the city, take some time to learn your way around."
"The street is like the playground when you're a kid, and you have to learn your way around," she said.
You have make new friends and learn your way around.
I've discovered that a comparison of two restaurants is a useful way to learn your way around a new cuisine.
If you decide to install iOS 7, as you learn your way around the new system you'll stumble across all kinds of handy features and techniques.
Once you learn your way around the primary city-run sites, although they can be frustrating to navigate at times, you are likely to appreciate the more complete access to information they can give you.
Now, the point of Politifact and other news-org fact-check things is supposed to be to do this work for readers, so that you don't have to learn your way around labor-force or trade or crime or whatever statistics every time you have doubts about a political claim.
Learning your way around is no picnic either.
Or perhaps you're learning your way around a new piece of industrial equipment; the app could show safety mechanisms, charging ports, disassembly instructions, all that stuff.
Outside of a few phones which had screenshot functionality hacked in by the manufacturers, nabbing a screen grab on Android generally entailed installing a massive SDK onto your computer and learning your way around the tools.
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