Well, to be honest, yes, it does cost money to buy and download songs online but it can also be free if you'd wish to make it free via the use of online mp3 converters which are known to be quite illegal on behalf of the copy-righting laws. If I were you, I'd just go and do it the safe way, buy the music and download it from iTunes. That way you're sending credit to the artist who own that particular song. But, to be honest, it actually depends what you specifally mean by asking "Do songs cost money on mp3 players" since we don't really know what mp3 player you're on about, but yes, songs do cost money.
Hope this helped? (: