The best way to master the game of golf is to begin closest to the hole to farthest away from short putts to the full swing.
The reason is simple: the short game is a microcosm of the full swing. In many cases, the skills utilized in short game shots the same as those used in the full swing, just on a smaller scale.
Closer to the hole, the swing is at its shortest and least complex: a smaller motion made using fewer body parts and thus less to coordinate. The full swing is the most complex, a much larger motion with many more moving parts at a higher rate of speed. Trying to master the full swing without first mastering the short game is like trying to gallop on a horse before you’ve learned to climb on.
A corollary of this truth is that many problems in the full swing can be spotted within the short game and vice versa. For example, if you come out of posture in your full swing, you will tend to do the same in your short game and will probably miss a fair number of putts. If you tend to reverse-pivot in your full swing, chances are you do this in your pitch shots as well. Improving your short game will not only make you a better scorer overall, but it will have positive repercussions throughout your game. As you work on and master your short-game shots, you will not only iron out the problems you are having in these particular shots, but your full swing will likewise benefit.
You will also develop feelings you need to understand and incorporate (e.g., what a proper pivot feels like) while working through your short game that you should also feel in your full swing.
This is not to say you won’t be using both short game shots and full swing as you play a full round but during practice time, which you must incorporate into your golf time to develop a solid game, the best way is to start small and work up from there.