What I would like to know is whether I should use System.currentTimeMillis() or System.nanoTime()<
As others have said, currentTimeMillis is clock time, which changes due to daylight saving time, users changing the time settings, leap seconds, and internet time sync. If your app depends on monotonically increasing elapsed time values, you might prefer nanoTime instead.
You might think that the players won't be fiddling with the time settings during game play, and maybe you'd be right. But don't underestimate the disruption due to internet time sync, or perhaps remote desktop users. The nanoTime API is immune to this kind of disruption.
If you want to use clock time, but avoid discontinuities due to internet time sync, you might consider an NTP client such as Meinberg, which "tunes" the clock rate to zero it in, instead of just resetting the clock periodically.
I speak from personal experience. In a weather application that I developed, I was getting randomly occurring wind speed spikes. It took a while for me to realize that my timebase was being disrupted by the behavior of clock time on a typical PC. All my problems disappeared when I started using nanoTime. Consistency (monotonicity) was more important to my application than raw precision or absolute accuracy.