Actually if both computers are networked using the wireless router, that would be a good option, especially if you have a shortcut to each computer's desktop from the other computer.
An easy way to transfer files is to use a USB flash memory drive (also called a "thumb drive", "jump drive", "pen drive", or "USB stick"). Plug it into the computer and right-click on the icon for the PAF data file, and select Send To (what ever your system calls the flash drive). Now click the the green arrow in your system tray (near where your computer displays the date and time), and get permission from Windows to remove the flash drive. Unplug it, then plug it into the other computer and copy the file to whatever location you want. You can do it in less time than it takes to describe the process. If you don't have a flash drive, you need to get one. One that costs $10 will have a greater capacity than a CD or a mountain of floppies.
CDs can be tricky. Most people I know have given up on buying CD-RWs. The common CDR is cheap and less trouble. CD-RWs need to be formatted before use because different software formats them differently, which means that other software may not be able to read them. On the other hand, CDRs come pre-formatted and so anything can read a CDR. Your experience leads me to suspect that you are trying to copy the PAF data files to an unformatted CD-RW or one that is formatted in a way that this computer cannot read. If it is a CDR, just drag-and-drop the file on the icon for the CD drive, or use copy-and-paste to paste the file's icon on the drive's icon. Then follow the instructions in the pop-up to write the file to the CD. (I have heard there is a trick that will allow PAF to save directly to a CD, but I don't know the trick.) One limitation of CDRs is that a second file cannot be saved in the same location with exactly the same name as an existing file. This is not a big problem. If you already have a file named genealogy.paf, you can name the new file genealogy2.paf if you want to save it in the same location.
Another popular option for small files (10 MB or less) is to send it from one computer as an email attachment, and then open the email on the other computer and save the attachment.
This is probably a ton of more information than you wanted, but you ask an important question and I hope that some of the info will be helpful to you or other members of this forum.