When you've finished working with a disk, be sure to unmount it, either via the graphical interface or with the umount command. If you forget, the disk may be corrupted. If the umount fails, make sure that none of your windows are accessing the removable media's directory and try again.
If you're using removable media to transfer files from another computer which is connected to the Internet, you might consider learning about ftp instead (or even scp, which is more secure but less user-friendly).
Writing to a recordable CD or DVD is slightly trickier than working with memory sticks, because you cannot simply copy files but must run a program to burn an image. On the new computers, the graphical CD/DVD creator will start automatically when you insert a blank CD or DVD.