Quickie review because I'm procrastinating on something, but...
First off, I'm just wondering. Are you typing this up in the reply to thread box? If so, you'll want to try using a word processor (Microsoft Word, OpenOffice, whatever you've got on hand along those lines) instead. The reason why is because you don't want to rush yourself, and you want to have an easy time proofreading. For the first one, if you're writing in the reply to thread box, you'll probably be tempted to write an entire chapter in one sitting instead of take your time with it because that's all you've got, right? One sitting?
As for the other one, writing in a word processor gives you useful tools like the spell check and the ability to save. That way, you can save your work and come back to it later when you're in the right mindset to proofread (i.e., less likely to be biased about your own work), and you can run your chapter through a quick spell check to dust off any obvious spelling errors.
Mostly, though, you'll want a word processor for the ability to save your work without having to post it in order to avoid rushing through a chapter.
That being said, let's talk about the chapter proper.
Couple of things to note:
First off, try to avoid the exclamation point in your narration. That causes your narrator to have more of a voice than you probably want. In other words, your narrator (if you're writing in third person POV) is usually a third party looking in on what's going on. They're not taking part in the action directly, so they really don't have a particular opinion on what's going on. The exclamation point makes it sound like they're experiencing the character's emotions when they really shouldn't be, in other words.
Second, it's not too bad of a start grammatically, but in terms of description, it might help you to add in something along the lines of describing what Zack looks like. Right now, all we can picture with some kind of detail is the arrangement of the room, but we don't really know much else. This is going to make things difficult for the reader to really get a good mental image going.
One way to start with description is by threading in phrases here and there to show the reader what a character looks like without going into too much infodumping. In other words, have descriptions connected with the actions they're taking. Let me give you an example of what I mean. (It's an example in part because I really do encourage you to experiment with it, and anyway, I'm not really sure what Zack actually looks like at this point.)
Slipping out of his bed, he stumbled across the room to one of the wooden bookcases. Colorful spines were jammed on every shelf except the top one, which was crammed with odds, ends, and a single plastic alarm clock that seemed to have been ringing for quite awhile. Picking it up, he squinted at the time. Then, he paused as the clock fell out of his hands and landed with a thump on the hardwood floor.
"Oh no," he whispered. "The Trainer Academy's entrance exam! I'm gonna be late!"
I could proceed to show you more, but you get the idea. Notice how the reader can sort of get a picture of Zack in their minds from reading all that? We can even sort of see how groggy he is as he stumbles around and squints and tries to tame his bed hair. That's the sort of thing you want to aim for: giving the reader a good look at what's going on and what things look like by describing things as the character goes. If you can get them to picture what's going on and what your world looks like, you'll be more likely to draw them in because they'll be able to feel like they can submerge themselves in your world.
Also, like I said, you don't actually have to use the story beginning I've written in the spoilers. That's just an example. Try playing around with details and actions to see what you can come up with yourself.
Whenever you write dialogue, you actually have to start a new paragraph every time the speaker changes. This is because it's a lot like starting a new topic. (You know what the teachers say in school. It's sort of like that.) So, instead of the above, what you'll really want is something like this:
“Argh!” he said as he hurried to get on his clothes and ran downstairs to hastily eat his breakfast.
“Good morning, dear,” his mother called to her son.
“Bye, Mom,” he said absentmindedly as he ran out the door holding his bag with his shirt on backwards, and his pants half on.
(This also makes it easier to read because your audience will tend to read each line as belonging to the first speaker in the paragraph. In other words, they'll think that Zack is the one saying the "good morning, dear" line until they get to the part about his mother. You want to make things clear at the very start, so your readers won't stumble, you know?)
'Course, there's a few other bits here and there to polish off. First off, you don't have to capitalize the word "morning." It's a common noun, not a proper one.
After that, it feels like you're rushing a bit through these actions. Zack comes down and eats his breakfast, but you don't really go into too much detail about what he's grabbing. That seems a little insignificant, I know, but it makes your narration seem like it's on fast forward. If he's in a rush, you could just skip the part about breakfast or give him something quick and easy to grab and eat, either right there or on the way. Either way, you'll want to avoid vague actions in your narration to avoid that fast-forward feeling, if that makes sense.
Strangely, you do family titles correctly in the quotes, so I'm thinking you don't really need to be told this. However, don't capitalize a family name (like "mother") unless you can replace it with a given name. So, you have it right with "his father" because you can't replace "father" with, for example, John. Likewise, you have it right with "where's Brother going" because you can replace "Brother" with "Zack" and still have the sentence make sense. You can't really do the same with "his mother" without implying something a little awkward.
You'll probably want to put this in its own paragraph as well, just because of the fact that because Zack's sister was the last one speaking, you're actually implying that she's also the one sighing here. Unless that's what you meant. At that point, sure, leave it as-is.
In any case, yeah, it's just an issue of paragraphing with dialogue, like I said above. To avoid having a reader stumble here and there, you'll want to start new paragraphs whenever a speaker changes. Likewise, if you've got a sentence of pure action (as in, someone is doing something) and no dialogue, you'll want to be careful with pronouns. Specify if you can or simply start a new paragraph if the action isn't done by the same person as the speaker.
This would probably be a better example of the fast-forward feeling I was talking about than the breakfast thing I pointed out earlier. Here, Chicole isn't exactly near Vientown. Sure, they're right next to each other on a map, but they're also separated by forest. Here, the narration makes it look like he ran through the streets for five minutes and then ended up in Vientown.
Beyond that, we don't really catch too much in terms of description. We don't know what kind of day it was or what kinds of other people he spotted (including potential classmates). There's no real interaction between Zack and his neighbors and friends as he rushes by (not even a farewell or good luck), so we can't really get a good handle on how he interacts with people or what his personality is like. (He could, for example, be a Paul-like character who gives everyone the cold shoulder, or he could be warm and Ash-like and say farewell to his friends as he passes them.) We don't even really get to see his thoughts, either here or when he sees the Trainer Academy for the first time.
In other words, do your best to slow down. If it takes a character one sentence to go from one town to another, that might be a good sign that you could fit a lot more in the space in between. Show us what Zack is seeing, and show us how he interacts with the world around him. That way, we can get a good image of what this character is like, and we'll be able to connect to both him and his world a bit more easily. In other words, it'll help us to really feel something for the world you're trying to present to us.
Overall, I wouldn't say it's absolutely terrible (i.e., don't get discouraged by how long this review actually turned out to be), but it could definitely be better if you give yourself a bit more time to write down your thoughts. Remember, the audience can't see your world until you show it to us by showing us what it looks like and how your characters interact with it. If you can pull both of those off, you'll be able to do a lot more with that plot you've got in mind.