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The golf swing plane is basically the path the club swings on throughout the backswing, downswing, and even the follow through. Many golf instructors talk about the backswing and downswing relative to the swing plane.
The golf club swings in a circle and it's not swinging straight up vertically or completely horizontal, it's a combination of the two. And what we want is the right combination of how much the club should swing up and how much it should swing around the body.
Having a good swing plane and swing path starts with good posture in the setup. If you are in a correct athletic stance with good angles, the club will automatically swing on a good plane or path in the backswing.
Keeping the body in the correct posture in the swing will also lead to a better swing plane and swing path. When the body moves out of position or out of the correct posture, the swing path of the club automatically changes.
Given that you are keeping your body in proper position during the backswing, if you move the club straight back to the point where it's parallel with the ground, it should also be parallel with your target line. The clubhead should be in line with your hands at this point also.
Once you've gotten the first part of the backswing, then it's simply a matter of swinging the club somewhere close to over your right shoulder (for the right handed golfer) to complete the backswing.
If you haven't manipulated the club or made extra movements with the body, then you're basically going to have swung your club 'on plane' in the backswing.
From a good position at the top of the backswing, it's then simple to make a correct downswing by trying to swing down the same path or plane you just swung up on in the backswing.
It's also a good habit to keep your follow through plane or path mirroring your backswing/downswing. So, feel like you are swinging your club towards your target after impact and then try to swing it over the left shoulder as you complete your follow through.