What are the key training areas (Openings, Middlegames, Tactics, Strategies, Endgame, Studies, etc) and tools (books, computer engines, databases, etc) in which players of different levels of playing strength (for instance 1800, 2100, 2400, 2500, 2600, Elo rating etc), should focus on in order to improve their chess to the next level of understanding and playing strength?
First, this outline is more for the average player, who still may be very decent, but is not likely to be become a professional.
Beginners up to 1800, it is mostly about tactics: Tactics, tactics, and more tactics. I also found that having a very strong dose of endgames, even when I got my first rating of 1036 USCF back in 1980, has stuck with me all the way through being a Master. I still love Fine's "Basic Chess Endings" (any mistakes in it as far overshadowed by the principles that it puts forth).
Around 1800, people start getting more clued in on positional play, and start picking up more on that. I still love Ludek Pachman's trilogy: "Complete Chess Strategy". That was the first time I started to see real plans enter into my games, and I still credit that trilogy with making me a Master. For the first time, I started to really grasp pawn play.
Also, around 1800, I also think that players need to understand HOW to play opening pawn structures instead of focusing on memorizing openings that when their opponent deviates, they are lost and on their own. For this, books like Soltis' "Pawn Structure Chess" (a really bad book if you are looking at it with a computer, but the underlying principles are very strong), and Mauricio Flores Ríos' book "Chess Structures: A Grandmaster Guide" is a fantastic book that expounds upon what was in the Soltis book, and does it a lot better, but also at a higher level.
Starting at 2000, assuming you have a shot at 2200 and higher, you are going to start needing specific openings, and memorization as you face stronger competition. All along, you should have been still studying more and more endgame material as you will also need that when you faces "monsters".
Over 2200, you are talking about in-depth refinement of what you have learned. A lot of people buy the Dvoretsky books before they are ready, and now is the time that you are just getting started to be ready for those. They will help with small details, and a lot of it above 2200 is analyzing better and better, so not only will his books refined and expand your knowledge, they are hand-picked difficult problems that need a lot of deep analysis.
Once at 2400, if you have not already had a professional coach, you probably will need one to really gain a lot more strength. In addition, if you have a kid, who is super talented, and clearly going places, it is also probably a good idea for them to have a professional coach at a lower rating, and younger age to get the most benefit.