"...all cd players read cd-rw? ..." No, it depends what players are they, for example PC drive does it but not all standalone players support CD-RW, especially the older.
Better use CD-R than CD-RW. Get a good brand as Verbatim or Taiyo Yuden. If you also use Nero it would be better.
Sorry, my english is limited, can you tell me what PC system you have and are you trying to tranfer the contents of your CD-RW to CD-R?