If a character's race allows a skill training (like the dwarf's Children of Grungni), does the character get that training for free or does it only allow to train that skill using creation points even if it is not her career skill?

It merely allows them the ability to train the skill if they want, even if it isn't a career skill. They do not get it trained for free.

Not to step on Dvang's toes but its been clarified in the Player's Guide. Quote from an example:

"... so Chris only invests 1 creation point in skills. This lets him train 2 skills, plus 1 skill for his racial ability."

plus all the racial traits say ...

"Before investing any creation points, a XXX may train one of the following skills for free – XXX, XXX or XXX"

Ah, another thing "clarified" in the PG. I stand corrected with the advent of the PG. I finally picked it up today, so I need to find time to read it.

Thanks for the answers!