A fellow gryphon fan! I'm in the middle of writing a novel on gryphons, too.
Caveat: I'm not a published author. So take this for what it's worth.
I think you have a good start and you can see the start of a plot. I use a dictionary, thesaurus, and a style guide (Strunk & White's) to help with grammar, proper word usage, and punctuation. The thesaurus in particular is incredibly helpful for finding different synonyms so I don't sound repetitive.

Second, I read my chapters out loud (usually to my kids). For some reason, I can catch the grammar mistakes better that way.

Third, I always have to watch to make sure I 'show, don't tell', and to use lots of description and action verbs. For instance, instead of saying "He was happy", which is _telling_ us, you can say, "He jumped up and down clapping, a wide smile covering half his face". That skill takes awhile to develop.

Use all your senses. Not only does the girl see something in that gryphon nest, I bet she can feel the rocks under her hands, smell the clean mountain air and/or the nasty odor of fresh gryphon guano, and can hear the creature give her a warning growl.

I've found books on writing to be extremely helpful.
This is the textbook I used in the college course, Creative writing, that was very useful: The Practice of Creative Writing: A Guide for Students by Heather Sellers

I also have read a variety of other books: Writing the Breakout Novel Workbook, How to Write a Damn Good Novel II, and Anne LaMott's "Bird by Bird". There are many other great books out there.

Make an outline of where your novel is going to go, from start to finish. Sometimes the outline will change, but you'll at least have a starting framework.

You don't have to write in order. Did I start my book with chapter 1? Yes. However, one day I was rather stuck, and another scene kept popping in my head, and the characters wanted me to write them that day (this is not a spiritual namby-pamby thing, btw--once you get to know the characters, they start doing things that you want to write down). So, I wrote out the scene that they were in. I liked it, and on top of that, it gave me a plot idea to use at the point where I'd gotten 'stuck' .

Do a little writing every day. The more we write, the better we get at writing.

My issue with your chapters is not the content, but the manner they're written in. There's a lot of redundancy, unnecessary commas and there are several areas that would flow a lot better if you simply used different words.

The flow of the story needs work, and the whole thing needs polishing and refining.

To be perfectly blunt, these few chapters you've posted are the equivalent of crude oil.

nice to meet another author (even unpublished for now) and thanks for the advice, though i have a few things to say:

i've had my family (mom and sister) read my story, and had their help on the matter

i do my best for the show part, but it still takes a bit of work... as for the descriptive element, i'm aiming for a basic style that will allow the readers to imagine the scenery for themselves, and not force them to accept what i'm describing

the senses, i'm still working on that... and another thing, i can't smell anything in real life (except for extremely strong smells), so that's why my character can't smell either

i had the entire outline of the book done before i ever started writing these chapters and also, writing in order is easiest for me (not sure why, but i guess that's my style)

but in the end, what you're reading right now is still just a rough draft, and not what i'm gonna present to the publishers (but still, thanks for the advice)

I agree with BoredZero. You should leave more to the imagination, or perhaps describe things in more abstract language. For instance, your first paragraph:

You keep the feel of the paragraph in fewer sentences and make the reader want to know what happens to both the asteroid and Earth. Is it Earth now, in the past, the future? Does it survive?

You do like commas, so after a paragraph like that you may want to stick to concise exposition. I'm sure you'll get a feel for it, but in my opinion you should never explain everything to the reader. The wonderful thing about the brain is that it will fill in "missing" information on its own.