How does one go about doing this?
Opera will grab the plugin from /usr/lib/flashplugin-installer/libflashplayer.so, which is the default location for the flash shared object installed by flashplugin-installer package. So make sure you have both.
If you don't have them, then install the package with:
If you have flashplugin-installer package and the file mentioned above, but it still doesn't work, then you could try to copy the plugin shared object to ~/.opera/plugins with the following command:
Also check if you don't have any old version being detected by Opera and disable it. You can do that from opera:plugins page.
For other video types you will need gecko-mediaplayer, which is the only media plugin that works with Opera.