Nero has a trial version, yes. Use that and see if you are able to do it. If you wanna keep Nero, buy it after the trial is up, or find a way of acquiring it illegally, as mentioned with the use of "torrents" (e.g., distributed downloading).
I've been using Nero ever since I had a burner (I've had four thus far), so Nero does get the job done.
I wouldn't look into Winamp or WM player as a solution for burning MP3 CDs, as they're only media players. That may seem odd, as MP3 is a form of media, but an MP3 CD is no different than a CD with "files" on it.
For most users, there are two types of CDs you can make: an audio CD (e.g., playable in any CD player) and a data CD (e.g., there's files on the CD, accessible like a file on your hard drive, for example). When you make an audio CD from a collection of MP3s, the burning program converts the MP3 into a mode readable by standard CD players. When you make an MP3 CD, the burning program does not make any conversions & just throws the file on the CD; it's up to the CD drive/host to be able to decode what's in the files.
As far as if a CD-R or a CD-RW will work, yes you can make an MP3 CD that is either on a recordable or rewritable disc. However, the total workability depends on three factors: 1) Can your burner burn to CD-RWs (most burners on the market do), 2) Can the end-user's CD drive read CD-Rs, and 3) Can the end-user's CD drive read CD-RWs? Also, some people have claimed that the actual media brand also plays a role, however I have not had any instances of not being able to use one particular brand.
So, in conclusion, just download nero.