M4A files are Apple's answer to Microsoft's WMA format. Conveniently, neither can see or use each other's format.
The saving grace here though, is that both can utilise the MP3 format, including your iPod.
Your best option is in iTunes to go to Edit > Preferences and then select the Advanced followed by the Importing tab. Make sure the format selected is MP3.
Once you've done this, scroll through your library and find all of the M4A format files you wish to convert (if you right-click on the view columns there should be an option that displays the format of the file to make it easier).
Once you've gathered them together and selected them, right-click and select the option 'Convert to MP3'. The process should then double all of the entries, creating an MP3 equivalent for all M4A files selected. After this, you can select again all of the M4A files and delete them.
Also, despite being unable (unwilling) to play WMA files, iTunes will happily convert them to the default import file format, which we changed earlier to MP3.
Try dragging and dropping your WMA files into iTunes to see if they'll convert.
Once everything is in MP3 format, you'll be able to listen to your Music in Media Player as well as on your iPod.