I have a brand new Lenovo G580 i3 6/1TB laptop. When I watch YouTube videos the quality is very poor (especially when in full screen mode).

Do I need an updated video driver?

The video driver is probably not the issue in this case.  As a first step,  check that the YouTube player isn't defaulting to a lower resolution.

Click on Settings in any YouTube video to see the default resolution (see screengrab below for guide).