Knowing theory

@29
OmegaDoom you have the highest training rating I had ever seen in lichess!
It's a complicated relationship between training and standard ratings.
Just few days ago i saw a 2300 player with a 1700 training's rating (after solving more than 1500 puzzles). I'm not sure what really determines the chess skill of a player but I think analysing your games with a strong chess engine is one of the best way to improve.
Also I'm agree that the speed of solving puzzles is important but it's not as important as you solve them correctly.

@31 In my opinion the best way to improve is to play lots of OTB (over the board) slow games and then analyze them, not with an engine but with a good teacher or by yourself, then check with the engine for tactics.
The reason why you shouldn't analyze with the engine imidiately is because engines play very often strange moves which make no sense for the human mind. As an average player consulting a tutor who will explain you the PLAN behind a move, why it is a good move and how to exploit your advantage is much more helpful and better than watching computer's += 0,75

@31
Ardavan74.
Solving a tactic correctly is not hard for me as long I have enough time. When I'm solving I don't rush and trying to see all possibilities. In a game the clock is ticking and I can't calculate everything. I analyze games(with engine) but don't see progress. I'm talking about speed of calculation because this is the most critical for games. In a game you usually don't have much time to think and it's a huge advantage if you can see a combination in seconds but your opponent only in minutes. Unfortunately the speed is more innate ability rather than trained.

@32
apostolis1
Engine can point out to huge oversights and here it's definitely useful for all levels.

@33 Engines are very good on tactical positions, but they are not good in positional ones. They play strange moves and analyzing positional positions with an engine doesn't help you improve your overall game and definately not your positional understanding.

@ #29 - Look at my training rating: 1789 and has been over 1800 after 144 puzzles. Now look at my classical = ~1300 and hasn't been over 1400 yet.

The disparity is due to time and inexperience. A lot of those training problems take me 15+ minutes to solve but in the end, I get them. In a game, w/ the clock running, it's obviously a horse of a much different color. I'm at least hoping that my ability to find and see 1800+ rated training problems means that eventually, at some point when I play enough games, my true rating will catch up to my overall knowledge and understanding of positions regardless of how much time is left on the clock.

Just to add something about my rating: It was, months ago, over 1450 on Chess.com. This rating here on Lichess was established over a short sample where other than the games this week, I had health problems, I was black every game, and my head/heart were just totally not into chess. So, my rating's going to go up here but still - There has always been a noticeable difference between my tactical ratings and my actual playing rating.

At least my mind and body are back in top form and I'm able to sit down and focus/enjoy the games again. As a result, yeah ...I'm starting to improve again.

A fair amount of good advice and thoughts in this thread.