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If found the expression "to learn to one's cost", which seems to be "learning a lesson at the expense of their money". Some examples:

Thousands of people - mostly internet-savvy men in their 20s or 30s - have learnt to their cost the legal and financial risks involved in file-sharing copyrighted music in large quantities.

Toyota have learnt to their cost that it is far better to be transparent and open when issues are identified than to try and cover up any faults.

Is this an idiom? Another definition of "to learn"?

Would it be correct "to learn to one's health"?

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You've got the sense of it, though the price need not be paid in money. For example:

Thousands of people - mostly internet-savvy men in their 20s or 30s - have learnt to their cost the legal and financial risks...

Learning a lesson about the dangers of file sharing may have cost them their freedom rather than just their money. For example, the proprietors of some notorious file sharing sites have been jailed over their conduct (1, 2). Anything which can replace X in cost someone X can be used in this way.

I wouldn't call this an idiom. It's perhaps a bit unusual, but the meaning follows directly from the definitions and grammar. Learn is used in the standard sense of learn a lesson.

Would it be correct "to learn to one's health"?

Yes, but it's extremely unusual and I've never seen this turn of phrase before. Native speakers will get the meaning but find it a strange thing to say. Here's a contrived example:

Upon drinking the supposedly poison brew, Socrates learnt, to his health, that it was actually the nectar of the gods, granting him eternal life and a place in Olympus.

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I'll try to be simple.

The expression to someone's cost means that the thing learned did not come easily. They had to face some difficulties or losses to learn it.

Oxforddictionaries describes the meaning of to someone's cost which is used in the sentences in concern-

to someone's cost - With loss or disadvantage to someone

Having said that those internet savvy men and Toyota learned whatsoever lessons after having faced difficulties or bad experiences.

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You are nearly correct in your guess - it doesn't necessarily mean 'learning a lesson at the expense of their money', it can mean at the expense of anything meaningful, for example, reputation.

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