Okay, here it seems we have an OT (Original Trainer) story. There are a couple pitfalls you should avoid if you choose this genre, but I'll get to those later.
A young girl was skipping away in a world of her own as she paused. The voice came again. She paused, mid-skip and turned on the spot.
Saying Hilda paused twice is a little redundant, unless she starts skipping again, which it doesn't say. I would get rid of one of the pauses.
Lillipup charged at Snivy, knocking him straight back to his trainer’s feet.
Hilda shouted commands at Snivy, and the attacks came back and forth.
Eventually, her first Pokémon battle was over.
The novelty still hadn’t worn off however, as she was still buzzing from the experience.
What happened? Here we get a battle, which can be very exciting, with nothing going on. We don't even know if the Lillipup was caught. I would flesh out this part of the story to make it a little more exciting.
"Pokémon are subject to the selfish commands of Trainers...
I would get rid of the quotation marks at the beginning of the quote, as this is already in the middle of someone speaking.
"That's right! We must liberate the Pokémon!
Then, and only then, will humans and Pokémon truly be equals.
Everyone, I end my words here today by imploring you to consider the relationship between people and Pokémon... and the correct way to proceed. We sincerely appreciate your attention." The man and all his minions walked off.
Some of the crowd began questioning each other and chatting.
Who is speaking here? Ghetsis was speaking earlier, so one could assume he is speaking, however, N was the last character mentioned, so I originally assumed it was him speaking. I would clarify that.
That's all I caught for grammar, but there is one formatting problem I seem to encounter again and again. You press the "enter" key a few too many times. Here's what I mean:
"That's right! We must liberate the Pokémon!
Then, and only then, will humans and Pokémon truly be equals.
Everyone, I end my words here today by imploring you to consider the relationship between people and Pokémon... and the correct way to proceed. We sincerely appreciate your attention." The man and all his minions walked off.
Some of the crowd began questioning each other and chatting.
Here, you started a new line after just about every sentence. That is unnecessary as starting a new line only comes in after starting dialogue, or starting a new paragraph. Even then there should be a space between them, as you seem to have down. There are exceptions to this, but this is a general rule. So the above dialogue would change to this:
"That's right! We must liberate the Pokémon! Then, and only then, will humans and Pokémon truly be equals. Everyone, I end my words here today by imploring you to consider the relationship between people and Pokémon... and the correct way to proceed. We sincerely appreciate your attention." The man and all his minions walked off.
Some of the crowd began questioning each other and chatting.
As far as OT stories go, there are a few things you should avoid doing. One thing that should be avoided is copying the plot of the game. This makes the story very predictable, which makes it boring. Another thing is having your character a "Godly" character. This means that everything goes right for your trainer and no real conflict takes place. If no conflict takes place, it makes for an incredibly dry story. However, if you plan to stray away from the plot of the game, and make your trainer encounter some extremely challenging challenges, and possibly lose (at first), then go right ahead.
I can't say much about the story in general, as it is an almost exact copy of the game, plot wise. Your writing is good, although you need to work on fleshing out some aspects, and I see no glaring problems. All in all, a pretty good start to what could be a very interesting story. I'll look forward to the next update and what twists you decide to put in there.