I'm not sure if it's a correct sentence (if trains can be used in the context of train rides in English).

This sounds off and fine to me at the same time. My English radar has never been that good anyways

I think it's referencing trains being made free for Ukrainians, so "for you" and "in the country" are there for a reason. I'd maybe reword it to "All trains across the country are free for you"

I would suggest something like:

All trains running in this country are free for you to use.

The extra clarification of "to use" is more idiomatic, i think.

> I suppose that instead of "to use" you could also say "to ride."

Yes, in US English (though not so much in UK English).

> I guess the Beatles were singing in American English for "I've got a ticket to ride."

(She's Got a) Ticket to Ride. Yeah.

Just to elaborate a bit: some over here might use "ride" transitively like this for trains and buses, but, usually, it would be more associated with situations in which the person riding has control of the means of transport, in the way you would ride a bike or a horse.

It depends on the city, so I won't insert such a sentence. Private companies offer free rides from the border, but I wouldn't say that all of them do.