Making the Stroke
The first thing to do is decide which of the two putting styles you want to use: the old-fashioned door-on-a-hinge stroke, in which the putter head opens going back, returns to square at the ball and closes after impact; or straight back and straight through. In practice, the putter head will always open and close on longer putts, but on short putts and as a way of thinking about putting, the straight back–straight through style has many advocates, including Lorne Roberts and top short-game instructor Dave Pelz.
The best way I know of discovering the style that best suits you is to spend a happy hour at a pro shop, testing putters. Limit yourself to three or four likely models and really pay attention to which feels best. Make relaxed putts, short and long; close your eyes and sense the clubhead moving. If you find that you prefer big face-balanced mallet-style putters, go with the straight back–straight through style of putting. If a heel-toe weighted blade seems more natural to you, go with the hinge style.
Then get to work on a practice green or a carpet stroking putts.Focus on making your swing path consistent. Sense that you take the putter back low and then hit smoothly through the ball with a slightly ascending blow.
At last year's Masters I asked Ben Crenshaw about the secret to great putting, and he said, "There's nothing like hitting the ball in the middle of the clubhead." I agree completely. Even a minute mis-hit gets the ball started off-line and with less than perfect speed. You should spend a good portion of your practice putting time simply trying to put the best possible strike on the ball. Don't hit to a hole. Try to make each putt smoother, sweeter, softer, purer and better sounding than the last. I promise that putting with those objectives will make a big difference.
Which Hand Leads?
There is no absolute here, but for many the left hand initiates the backswing and guides the stroke by controlling the mechanical elements. I often think of the left hand as traveling along a waist-high rail parallel to the target line, with the back of the hand facing the target.
The right hand, in this scheme, provides feel and controls distance, and you may sense that it assumes the lead as the stroke begins its forward motion. To have any chance of developing good touch, however, you must avoid tensing the hands and forearms. You want to swing like a pendulum, but a human pendulum, not a grandfather clock made of steel and mahogany. Good putters need softness in the hands and a sense of lag in the transition between the back and forward strokes.
After the Ball Is Gone
The best putters hit through the ball with relaxed but forceful strokes, and their putter heads stop and hover for a beat or two six inches to a foot in front of where the ball was. By contrast, the putter heads of bad putters tend to recoil instantly, indicating tension. Work on holding that follow-through in a natural, easy manner.
I'm also an advocate of keeping your eyes focused on the ground after putting—ideally until you hear the ball rattle in the cup. I got some grief for doing this in my prime, but I am not alone: Gary Player and Tiger Woods have also done this at times. Better to overkill than to underkill, I always say. Keeping your eyes glued to the ground is terrific discipline.