Here are some examples:
Prefixes:
* Un-: Unconcerned (meaning lacking concern or worry)
* Non-: Nonchalant (meaning casually unconcerned or indifferent)
* Care-: Careless (meaning lacking care or attention)
Suffixes:
* -ly: Carefreely (adverb form)
* -ness: Carefreeness (noun form, referring to the state of being carefree)
Keep in mind that these words have slightly different nuances than "carefree," but they can be used in similar contexts.