" Alright? Oh, playing a bit of chess? Well, err , what you wanna do there is sorta like checkmate him an' that but err don't , like, get..get checkmated yourself. Sorted" - Garry Kasparov
Memorable Sayings for Teaching

'Don't advance a pawn unless it's correct to do so.'
'Don't trade pieces unless it's correct to do so.'

Chess is the fairest of all games ( Isaac B. Singer )
Chess is fun ( Luke McShane )
Chess, like love, is infectious at any age ( Salo Flohr )
Chess is life in miniature ( Kasparov )
Chess excludes chance ( Réti )
Chess is work ( Walter Browne )
Chess is international ( Edward Lasker )
Not specifically to help young players to learn the principles of chess, but use one line per week as a motto and the youngsters will understand, especially the third one...

I'm not a coach, but I know people who get frustrated by losing so I find this one always helps.
"In chess, winning comes from experience. Experience comes from losing."

'A knight on the rim, guarantees the win'
'A bishop behind pawns, feels fuzzy and warm.'
'Keep a rook a mile, away from an open file.'

There's a great chess thinker who calls a rook on the 2nd/7th a 'piggy', presumably because it can gobble the pawns like a pig. I thought that was pretty good.

"Chess: don't play it because once you start you can never stop." is a good one to tell youngsters
Blackburne said something along these lines:
Joseph Henry Blackburne on Addiction and Chess
Edward Winter's exceptional Chess Notes column on 7 January, "Chess and Alcohol," carried an image of an 1895 republication of an interview with Joseph Henry Blackburne. The article was published first in the Daily Chronicle and then in Chess Player's Chronicle; Winter reproduces it.
The reporter asked Blackburne whether chess is "the intellectual pastime that some people declare," whether it has a place in schools, and whether perhaps it might even serve as a substitute for geometry. Although the question seems a bit over the top, Blackburne's answer serves a cautionary footnote to the efforts of many (including me) who push chess into the school curriculum. The reporter might have asked whether it could supplement or precede the study of Euclid (original works in geometry), rather than replace such study. Would Blackburne's answer have differed? We cannot know. But the truth of his remarks ring true in any case, at least they do when we consider the widespread ailment known as an online blitz addiction.
Blackburne said, in part:
Decidedly not. I know a lot of people who hold the view that Chess is an excellent means of training the mind in logic and shrewd calculation, prevision, and caution. But I don't find these qualities reflected in the lives of Chess Players. They are just as fallible, and as foolish if you like, as other folk who don't know a Rook from a Pawn. But even if it were a form of mental discipline—which I take leave to doubt—I should still object to it on the ground of its fatal fascination. Chess is a kind of mental alcohol. It inebriates the man who plays it constantly. He lives in a chess atmosphere, and his dreams are of gambits and end games. I have known many an able man ruined by Chess. The game has charmed him, and as a consequence he has given up everything to the charmer. No; unless a man has supreme self-control it is better that he should not learn to play Chess.
It has been years since I've read Alexander Cockburn, Idle Passion: Chess and the Dance of Death (1974), a book written in the wake of the Fischer boom in the United States. As I recall, however, Cockburn's argument against chess seems almost a book length meditation on this brief statement by Blackburne.
Chess is intoxicating, blitz especially so.
http://chessskill.blogspot.com/2009/01/blitz-addiction.html
A lot of coaches for young players teach easy sayings to help young players learn elementary principles of rules and strategy.
Touch a piece, move a piece
Knights before bishops
If you see a good move, look for a better one
I came up with two yesterday. I don't like both of them. Suggestions are welcome.
The king is old and needs a blanket of pawns.
The runner (bishop) needs to breathe.