Actually, there's a logical progression for improving at the game. I'm sure it's different for different people, but for me it was:
Learn to 2-putt from anywhere. Obviously I don't mean huge distances or complex multi-breaks, for 2-putt the typical green from 50 feet on in. If you do that, you'll stumble across some 1-putts.
Hone your approach shots. Once you hone your close approach shots, you'll get more 1-putts to make up for the missed approach. When you've improved your long approach shots, you'll have longer putts for a while until you've tightened up your accuracy.
Keep your drives in the fairway. First cut rough isn't much of a penalty, but anything else is.
Sad, but true, you can buy improved performance. Better precision and more forgiveness in your clubs and balls. Clubs and balls with more spin stop faster and the carry is a lot more controllable than the carry-plus-roll.
Just my viewpoint.