If your music is displayed in iTunes as if it should play but then skips until the end of the playlist in rapid succession and everything worked fine before updating to Windows 10, chances are that your sound card driver is Realtek High Definition Audio and its version is something below 6.0.1.7544.
You may check and update directly from Windows audio driver settings. For that open Sound properties, find the device which you are currently using, on the general tab go to controller information / properties, open tab Driver and check the version. If its below 6.0.1.7544, click on the button Update Driver, select update online and wait for the magic to happen. Afterwards, restart your computer (you may try with just restarting iTunes). So, all I can say, this worked for me perfectly.
Updating the driver version this way may also help for above problem with other vendors besides Realtek.
BTW: This solution works also for Spotify. ;-)
BTW2: In general it proved very safe to ignore any "tips" like logging out/in again, reinstalling iTunes oder restarting your computer without prior doing anything else. It just won't work. Anyhow, these crap keeps turning up since - well, ever. It's a pity...