Most golf clubs have some old boy who putts very nicely indeed using an unorthodox stroke. He may chop the putt from outside to in, scooting the ball with a little sidespin—and darned if the thing doesn't drop in the hole more times than not. If he can make that same motion every time, fine; I'm all for individuality. But the more faults you have in your stroke, the more apt it is to break down, either under pressure or for no reason at all. So you're better off developing a sound, fundamental stroke: one that quickly gets the ball rolling as purely as possible without any sidespin.
The goal of the putting stroke, in brief, is to create a nice triangle from your shoulders down your arms and forearms to the hands, and to use that triangle to move the shaft back and forth in something close to a pendulum motion. The pendulum is the optimum motion we have on this planet for transferring energy, but when the body and shaft get out of sync, the pendulum breaks down. The shoulders control the motion of the triangle. You want them to rock rhythmically as you putt, up and down, as in the Umbrella Drill. But for the shoulders to have the freedom and space to rock correctly, you need to first master a few other fundamentals.
You see all kinds of putting grips on Tour these days: the claw, the saw, the left-hand-low. It's good that golfers no longer feel stigmatized for using the grip that works best for them. But as a place to start, I recommend a light grip with the palms facing one another and the left thumb running down the top of the shaft. If the shaft is in the fingers more than in the palms, you'll have a better feel and your forearms will tense up less. The firmer you can keep the left wrist, the better. Try hitting putts with just the left hand. You'll quickly discover which left-hand position feels most stable for you.
Stance and Posture
Again, it's all about finding a relaxed, comfortable position that gives your arms and shoulders room to operate. For most people that means flexing the knees slightly and bending forward from the waist with the spine as straight as feels natural. If you let your arms hang down loosely and then join the hands to form a grip, you'll get a sense of how much to bend forward. Ideally, you want the shaft of the putter and your forearms to form a straight line—that is, the forearms and shaft should be on the same plane. It's also extremely important that your shoulders be parallel to the target line. When people line up with their forearms or shoulders out of alignment, they have to compensate in the stroke to get the putter head back to square at impact, introducing all kinds of opportunities for mis-hits. Once you get used to aligning everything square, you will have a good standard to refer to when your putting gets a little cockeyed.
In addition, try to feel as if your elbows and upper body are connected. This will help synchronize the torso's rotation around the spine with the movement of the shoulders, arms and hands. As for ball position, in most cases it should be slightly forward of center, and your eyes should be over or slightly inside the ball. You can test this by dropping a second ball from your left eye and seeing where it lands.