Most YouTube content requires the Flash plugin and on occasion needs troubleshooting.
Open System Preferences > Flash Player then select the Storage tab. Click: Delete All
Now uninstall the Flash plugin then reinstall new > Troubleshoot Flash Player | Mac OS
From your Safari menu bar click Safari > Empty Cache
Now try a video.
BTW... if you have the ClickToFlash extension installed, that can prevent Flash based video from streaming. It can also be installed as a plugin in /Library/Internet-Plug-Ins.
Keep in mind, it's best to never reinstall the Flash plugin from any site but Adobe. Othewise you may invite malware on the hard drive.
Congrats on your new Mac !!