What am I allowed to do on the putting green?

Marking and lifting (Rule 13.1b): you may lift and clean your ball on the green, but you must mark its spot before lifting and replace it on the original spot.

Repairing damage (Rule 13.1c): you may repair damage on the green without penalty, including ball marks and spike marks, using your hand or foot, a ball-mark repair tool, a tee, a club or similar equipment. Don't unreasonably delay play, and don't go beyond restoring the green to its original condition (for example, creating a path to the hole). That's the general penalty (2 strokes in stroke play).

Flagstick (Rule 13.2a): you can putt with the flagstick left in the hole. If your ball hits it, there's no penalty and you play the ball as it lies. The exception is deliberately moving it off-centre to gain an advantage (general penalty).

Hitting another ball (Rule 11.1a): in stroke play, if your ball played from the green hits another ball at rest on the green, and both balls were on the green before your stroke, you get the general penalty (2 strokes). Ask for the other ball to be marked if it might be in the way.

Holing out: in a medal you must hole out on every hole (see the question on not holing out).