Setting up correctly is the first step to improving your putting. You want to get your eyes over the ball so you can see the line well. Let the arms hang down naturally and try to have light grip pressure. On a scale of 1 to 10 I recommend holding onto the putter grip at around a 4.
The putting stroke is like a pendulum and the putter should move back and through smoothly. Let the ball get in the way of the stroke, don't try to hit at the ball.
It can be helpful to think of a triangle with the hands, arms, and shoulders. Try and maintain the triangle as you make your backstroke and forward stroke with the putter. The shoulders are the main initiator of the motion, the hands and arms are hanging on and moving along with the shoulders.
Typically you want to go back and through about the same length. On shorter putts, it's okay to make the follow through slightly longer because you want to make sure you're accelerating towards the hole or down your line. Keeping the putter head low to the ground is important because lifting it up too high can put side spin on the ball and cause you to miss putts right or left.
The path of the stroke should be relatively straight back and straight through. For longer putts, there will be more of an arc to the stroke. Make sure the putter head comes back to impact in the same place it was in the setup position. A slight opening and closing of the putter face is good but doing this too much makes it harder to time correctly and repeat it. Keeping the putting stroke simple will help you be more consistent.