Log in to Youtube and click your profile button in the top right, go to settings. On the left, you will see a playback option. Go there and select the "Always choose the best quality..." and check off the "Always play HD on fullscreen (when available)".
Now, mind you, Youtube will almost always select 360p when you first load a video which really sucks. Even when you fullscreen it, it will often take up to a minute for it to automatically switch to the higher resolution. For Youtube viewing, I like to use an app called "MegaTube" where you can set it to always just load the HD video right from the get go. You can then also save any video you watch for future watching, such as for on the subway or bus where you don't have internet. It's pretty awesome.