According to dict.cc, there are three similar ways to say it: sorglos, sorgenlos and sorgenfrei. This doesn't include the other translations such as unbekümmert or unbeschwert, either! Then which one would I use, and what are the difference(s) between them?
Also, another question that someone could possibly answer: if I am met with multiple translations for one word like this, which one should I use? Sometimes I just choose one randomly, and a German speaker tells me that the sentence I am making sounds really strange, unlike one a native speaker would make.
As a commenter mentioned, here is the sentence I would like to translate:
The children smiled and looked happy and carefree.