To start this off, I will share a tip from Kotov's Think Like a Grandmaster book. Strong players tend to use their clock time, when it's their move, calculating variations. They use their opponent's time for positional considerations and taking a bird's-eye view of the position.
Chess Wisdom
I think Ruy Lopez advised playing when the opponent had drunk freely. Oh, and facing him into the sun as well.

Rodrigo (Ruy) Lopez de Segura was a clergyman!
"A passed pawn increase in strength as the number of pieces on the board diminishes." - Capablanca

Great advice so far! I remember reading that GM Kevin Spraggett used to train one hour doing tactics (focussing on speed) followed by one hour solving endgame studies (focussing on depth and accuracy of calculation) everyday.

I remember reading some advice by GM Conrad Holt in which he said that he went through Dvoretsky's Endgame Manual and knew all the positions in there by heart. His advice was to do the same.

Axel Smith, in his book, stresses the importance of making opening repertoire databases replete with your own annotations.

One method of training I came across recently was the 'Stepping Stones' method suggested in one of the latest books on improving one's Chess rating. Basically you try to calculate a bit when solving a problem, and then when you get to a point where you can't 'see' the position any more, you to visualize hard and picture it in your 'mind's eye;' then you try to go even further.

I think Ruy Lopez advised playing when the opponent had drunk freely. Oh, and facing him into the sun as well.
A historical note - that's actually from Liber de Ludo Aleae.

When playing blitz on a real board, there are some dirty tricks you can use. You can put the pieces in between two squares to throw off your opponent. You can also move only the pieces closest to the clock to save time.

I read somewhere
"Chess is a pool of water that elephants can bath in and gnats can drink from"
I am a good drinker.

Another Capablanca quotation to take to heart:
"You may learn much more from a game you lose than from a game you win. You will have to lose hundreds of games before becoming a good player."

Any time you acquire wisdom from playing chess, you need to ask, When did I lose that aspect of Common Sense?

Everytime it's your turn, the first question to ask yourself is, what is my opponent attacking or threatening? You should get into a habit of always asking yourself that. Over time you will see your opponent's threats instantly and won't have to consciously ask yourself that question everytime. I read this off of GM Spraggett's blog.

You have to play each tournament game like your life depends on it. I forgot which book I read this in.

You have to play each tournament game like your life depends on it. I forgot which book I read this in.
Come on, omar,
You have to read every book like your life depends on it.

Don't remember where this one comes from:(Paraphrased)
"Which rook to move to the open file? Weigh all the pros and cons of using either rook. Double check. Once you have decided which one to move, move the other one."

This one I've never seen applied to chess, but I think it could be:
I heard this idea when I was in University. Someone said it is a useful test when trying to prove a theory is correct. Instead of proving why it is correct, you try to prove why it is wrong. And, in the end, if your theory is correct, this process will help prove it.
When choosing between 2 difficult candidate moves, try to prove to yourself why each candidate move is the wrong move.
Let's use this post to share all the Chess wisdom that we come across, namely, tips on how to improve or perform better.