One of the most important fundamentals you should pay close attention to when learning how to putt better is the putting grip. Having a proper grip is of utmost importance in putting as it allows the hands to work together as a unit with the shoulders. By having a poor putter grip, however, you're setting yourself up for years of agony on the greens as it will likely lead to an overly wristy and handsy putting stroke that will lead to inconsistency and even the putting yips.
How to Grip the Putter
Gripping the putter properly starts in the lifelines of the palms. Rather than holding the putter grip in the fingers, you want the corners of the putter grip to run through the lifelines as this gets the shaft in alignment with the forearms as can be seen in the image below.
With my forearms in alignment with the putter shaft, the wrists, arms and shoulders can now all work as a unit rather than fighting against each other. This dramatically simplifies learning how to putt and will make you a much more consistent putter.
The Reverse Overlap Putting Grip
The reverse overlap is probably the most common putting grip that you will see used by pro golfers. The reverse overlap putting grip is when the index finger of the left hand drapes over the fingers of the right hand as it extends down the putter grip. This unifies the right and left hands on the putter grip so that one won't easily overtake the other.
Once you have the putter running through the lifelines of the palms and the reverse overlap, you will note that that your forearms will be in perfect alignment as well. This putting setup gives you the best chance becoming a great putter with the least amount of effort.
Now, watch the video on learning "How to Putt" and the putting grip: