I like this thread for some strange reason. x.o;;
As someone before me specifically stated, it is really based on the trainer, but can also be based upon the plot, the settings, the time, and the right capture. For example, I am currently writing an OT fanfiction and the main character will not capture his first Pokemon (besides his starter) until possibly after the second or third badge. It really varies upon what Pokemon they want to go after; a dragon wouldn't be obtained until you go hunting for such a long time, while a flying-type could be acquired at any particular time, since they are constantly found roaming the air and the skies. Some trainers might want to assemble their team as early as possible, and others might not. Some might want to have all six to train with and have a variety of choices, and some trainers might want to focus on each Pokemon individually until they build up enough strength where it'll be enough to win battles as they focus on a newly-captured Pokemon.
However, my personal preference is that it all comes natural; for a writer whose composing an OT fanfictional piece, let the Pokemon and the writing come to you. If you mention a wild Sneasel, that doesn't necessarily mean you should let little trainer Georgia capture it and be happy; when you feel it's the right time for your character to make another capture, then let it somehow flow into your story like how a young Skitty might get caught into an Ariados's spiderweb; quick and fast at first, but intelligent and thoughtful once trying to get out. Let things happen at a good pace, but when it is time for action, think things out before going on and continuing the journey.