Getting a kill with the tank should be fairly easy, you can use either the turret or the machine gun. Blowing up an enemy vehicle is the easiest option but don't forget the machine gun for taking out any enemy if they are on foot.
The car kill is a bit trickier. You can get on the back of a jeep while a team mate is driving for a much easier time in finding a enemy on foot, if you can find someone who will drive you :) Or you can try this by yourself by driving to an enemy flag to capture it, change seats to the machine gun and take them out as they respawn near you.
The plane kill, there is fortunately an easier way than flying those **** airplanes :) Find an air raid bunker and use it, aim your bomber plane over an enemy flag and fire, hopefully you should get a kill.
Once you get a kill with each of them the achievement will unlock.