I just got a PS4 yesterday but noticed that when I inserted my game disc (FF XIV) there was no option to install it to the HDD, does the console install the game automatically, as I like to install my games to the HDD?

Not sure if it is automatic, but it is mandatory (games run better off the HDD than they do off the Optical Disk)

See this

http://www.youtube.com/watch?v=u26iU5Ns_uY

From what I've found, the PS4 likes doing things discretely. If there's an update for a game you have it'll automatically download in the background, and I believe that it installs games to the HDD automatically when you insert them. If you can start and play the game, assume it has been installed.

Great thanks, the game installed really fast considering it's 39GB.