Some of these may not really be mantras but just things to remember, so here goes!
"The Button wins"
This is something i have heard thrown around, it is obviously not always the case but the reason why i have opened with it is because it is one of the most fundamental points to remember in Hold 'Em poker. Position is always important and just remember, having position on a player can sometimes make your cards almost irrelevant.
"Play the player, not the cards"
This 'saying' applies almost exclusively to No Limit Hold 'Em, especially at the higher stakes. In NLH, having a read on a player and understanding their personality and playing style can be just as important as the two cards you are dealt. For example if an aggressive player has position on you then you may be best to check a hand that you believe is the best and induce a bluff from them, you can then implement a coy check-raise over the top.
"Bet with the best, good draw to invest, fold the rest"
I find myself living by this at the poker table, and to some extent every player is when calculating their pot odds in any given hand. This mantra simply means that whenever you hold a strong hand, like top pair after the flop, you should bet/raise with it. Whenever you hold a flush or straight draw you should call, as long as the price of the call is right for your odds of making the hand.
"Weak players will make mistakes, don't try to force them into making more"
When facing a 'fish' who is unable of putting you on a hand and will over-value most of their hands, the best way to win is often to just stay away until you can develop the strongest hand and value bet them for all they are worth! This works because they are unlikely to fold their low pocket pair even when you have top pair top kicker when it comes to the river. The will also often rule out the possibility of you making that flush with 3 clubs on the board and assume that their Kings are best.