Go out of your way to do something

Yes! "To go out of your way" to do something, is a phrase that is used very often english.

It can mean "to work hard for something", like you said, but can also mean "to do something that you do not normally do".
Literally "go out of your way" or "do something in a different way" all with the intention to help others.

"I went out of my way to help the girl get home".
I did something different (or went a different direction) then I normally would go, to make sure I was able to help the girl get home.

In this sense, I do not think it make sense.

I believe you are thinking of the meaning similar to your original definition of "trying really hard to do something"

"the best thing we can do to learn another language is to go out of our way" this seems like you are trying to say "the best thing we can do to learn another language is to try very hard".

You need to specify more of what exactly you are "going out of your way to do".
This expression does not have a sense of "trying hard" unless you state this specifically.

example: "To learn a language you need to go out of your way to try hard"

But in my opinion the best way to express this idea would be:
"To learn a language you need to go out of your way to practise every day"
"To learn a language you need to go out of your way to put in effort"

As you can see the examples are more specific about what exactly you are "going out of your way" to do, which will help you achieve the goal of learning a language.

Hope this helps!

You are very welcome.

Yes I am Canadian