There is a lot of discussion on whether starting on 19x19 or 9x9 is preferable, the general consensus tends to 9x9.
Personally, I started on 19x19 but would have preferred 9x9.
As a beginner, you first need to understand the most basic melee fighting tactics (atari, ladder, snapback, basic life and death, etc).
Those can be learned on both big and small boards, but small boards avoid introducing complex strategic issues and make for faster games that are easier to review.
You probably should stick to small boards until you are familiar enough with the game to stop making "obvious" mistakes (crawling with a dead group on the first line, playing out ladders, stuff like that).
boardgames.stackexchange.com/users/60/mafu
If you have played go for a while, a full board can seem like a natural place to learn.
Obviously it is possible; So
that asks the question: "Why do so many people learn on a 9x9 board?"
and "Why does a 9x9 board exist at all?"
I have found, (past 10 years of teaching
Go), that a 9x9 is the right place to teach people.
There are lots of reasons, (that you will probably find in the other answers - "get through your first 100 games quickly"), but my top reason is fear.
19x19 lines looks like 18x18 squares to someone that has learnt to play, almost all other board games.
The mind can boggle at how complicated that must be;
A 9x9 looks like 8x8 squares, which is the same dimensions as some other games use, (though they are mostly tactic based games and Go adds strategy into the mix.)
So to sum up, some reasons to learn on a 9x9: Friendly size Get through teaching games quicker Learn the rules without the distraction of the deeper complexities, (the fun part), getting in the way.
Still fun Cheaper to make or buy Easier to store
boardgames.stackexchange.com/users/3288/Alexx Roche
A 1/4 board (11x11), is not a standard size , so I do not believe that any official setup for black handicap stones exists.
Some confusion exists on what a quarter board is though, as I have seen a 9x9 also called a quarter board.
The handicaps for that board are listed here .
boardgames.stackexchange.com/users/1873/user1873
a 9x9 is a board that is set for a battle not a war.
and that is ok.
There is a lot less centre area in a 9x9, which is good because there is no territory in the centre.
I find a 9x9 takes 15 minutes to pay,where as a 19x19 will take an hour or more.
A 13x13 is a interesting compromise.
A small war.
boardgames.stackexchange.com/users/6665/Lyndon White
Actually setting up the 'quarter board' is hard to do on a 19x19 board.
Most people use a small board specifically made for the size, or make their own.
It is possible to crop an area from the large board, but this can be inconvenient.
As for learning on one, I feel Pieter Geerkens comment is quite correct.
13x13 gives a more balanced vision of the game than 9x9.
19x19 is too large, 9x9 is too small, but good to introduce the rules with.
Once the player has the basic rules down 100% on 13x13 then 19x19 becomes more interesting.
After some years playing primarily 19x19, I've gone back to 9x9 and 13x13 to improve my life and death techniques and starting moves.
boardgames.stackexchange.com/users/6767/karmington
The basic board sizes are 9 x 9, 81 points or a "one-quarter" board, 13 X 13, or a "half" size board of 169 points, and 19 x 19, for a "full" board of 361 points.
Many full sized boards will have a 9 x 9, or 13
X 13, board printed on the back.
Otherwise, you could take paper and "block off the extra ten rows or columns of points above 9 x 9 on a full board.
I learned to play Go on a full-sized board and never looked back.
You might want to use a smaller board for the first five or ten games, just to learn the mechanics, but you'll want to transition to a full sized board as soon as possible so that you don't learn strategies that work on the smaller board but may be inappropriate for the full version.
boardgames.stackexchange.com/users/1344/Tom Au
As a beginner, I was handed a full go board. And, somehow figured it out.  A beginner friend wants to start "small" on something like 1/4 of the board.  Personally, I think that we should start with the entire board.
How does one set up a 1/4 board for Go – and is it a good way to start?
boardgames.stackexchange.com
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boardgames.stackexchange.com/questions/7771
boardgames.stackexchange.com/users/52/wdypdx22
It is possible to crop an area but it would be inconvenient. 9x9 is a board set for battle not war. A 1/4 board (11x11), is not a  standard size , so I do not believe that any official setup for black handicap stones exists.  19x19 is too large, 9x9 is too small, but good to introduce the rules with. A 9x9 is a board that is set for a battle not a war.   S2: There is a lot less centre area in a 9x9, which is good because there is no territory in the centre.  
It is also possible to crop an area from the large board, but this can be inconvenient.  
A 1/4 board (11x11), is not a  standard size , so I do not believe that any official setup for black handicap stones exists.  19x19 is too large, 9x9 is too small, but good to introduce the rules with. A 9x9 is a board that is set for a battle not a war.   S2: There is a lot less centre area in a 9x9, which is good because there is no territory in the centre.  
It is also possible to crop an area from the large board, but this can be inconvenient.  