From my experiences,improving skills takes time and patience, and a combination of those things that you listed. Personally, I find it very helpful to analyze tactics though....
Improving chess skills

Oddly enough playing japanese chess or shogi helped me get better. I suppose it is becuase shogi is more positional and most pieces cannot move backwards so it made me think alot more before moving, immensly improving my positional game.
Chess Books have also helped my opening alot and taught me alot more about getting ahead of opponents in development during the opening, the greater development the more damage you can cause to your opponent.

I think patience is. really. Just like most things you just have to play much, do your best and enjoy playing.
I have bought only one book on gambits once but I never really read it. I think this becomes more interesting once you specialize your game. Puzzle's are fun and useful but it's always one specific situation.
So I'd say just play and analyze your games

Analyzing your own games for common mistakes, and then customizing your training plan to address them. But for weaker players, knowing the basic mates (queen or rook) by heart, knowing basic opening principles, and then mostly studying basic tactics--especially counting problems (1, 2, 3, 4)--and then forks, pins, skewers, and discoveries is a great place to begin. Oops, is that more than one thing? ;)
What would you say is the best and the quickest way (one thing) to improve chess skills?
Puzzles, reading chess books, playing chess, analyzing games of professionals...?
What did you do to make your chess better and did it worked?
Hope to see some answers :)
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Oh.. and 1 more thing. Maybe you know.. How do GMs become GMs? Do they have some special training or something (or do thay all have trainers) or thay just do what we do? :)