Tons and tons of demos on PSN. Just start queuing them up.
infamous, Uncharted, LBP, Motorstorm, Gran Turismo, etc. Lots of franchises to try. Also Plus has been a really good deal lately, so it might be a good way to get exposed to a little of everything.
DC Universe Online is free and much fun, even my six year old boy loves it. Free Realms is also free to play, but I have not spent much time with it.
Also, Journey comes out today.
Now playing: TLoU PS4
Recently Finished: Watch Dogs, Southpark: Stick of Truth, Infamous: SS
PSN & XBL: Scuzzlebutt33