For backing up files on a regular basis onto a CD, I gather it is best to use a "CD-RW" type of CD. What would the problem be if a "CD-R" type of CD is used.

lol, there's no problem m8. You can re-use a cd-rw but not a cd-r.

Keep a couple on the go.
e.g.
Backup 1 on Disc 1
Backup 2 on Disc 2
Backup 3 on Disc 1
Backup 4 on Disc 2
etc
That way your backup always has backup.

I've always found CD-RW's extremely unreliable myself.

Also, my system needed a spindle full of CDR's for a full backup.

Since getting a USB2 external HDD I have never looked back, superb piece of kit.

I got the Freecom Classic 250Gb from Novatech.

Yep I've always forn cd-rw a pita, saying that I've just bought 10!!
I remember the bad old days when blanks of any kind were expensive. Now one can buy good quality blanks >£0.10 there's little point in using -rws.