In Kurdistan, public transport is rare although regular buses do link Zakho and Dohuk and cost about 2 USD. From Dohuk, shared taxis leave all day for Erbil and other cities. The road from Dohuk to Arbil goes south near Mosul, but does not leave Kurdish territory and is thus safe, although perhaps too close for comfort.
Shared Taxis might be the safest way to travel in Iraqi Kurdistan, as the drivers are not interested in leaving the province either.
By car
Driving at night may be a safer alternative to daytime driving, but a few rules to follow:
Avoid city centers. Although most Iraqis are asleep by midnight, the few that are awake are almost certainly up to no good.
Watch for the military. If you are out late at night and effectively trying to blend in with the locals, you could be mistaken for a hostile/troublemaker. At checkpoints, you will also be treated as a suspect, and until they decide you are not a target, you must conduct yourself carefully.
If you do encounter the military, ensure your lights are on, turn on your hazards/flashers, slow or pull over to the side of the road and follow any and all instructions given. If a stop sign, green laser, or any other signal is directed at you or in your general direction it is advisable to follow it, better to err on the side of caution than get shot at.