Peppermint pushed open the wooden door and entered her classroom. She gasped at its arrangement. The walls were painted in a deep, pink hue, which reflected against the lighter squares of the checked floor tiles. Dainty golden lanterns were held up by strings attached to the ceiling. Royal blue drapes hung over beautiful stained glass windows, which were glittering against the sun's reflection upon them. But what naturally caught Peppermint's attention the most, were the colourful books propped up on an antique wood bookshelf. Oh, think of the information in all those pages, waiting to be absorbed into one's mind!
She let her eyes wander over the rest of the room, and her sight came to rest upon the teacher's desk, and a tall pony next to it. She had a multi-colored mane with a pastel touch. It waved and swayed continuously. She noticed Peppermint staring at her, and smiled.
Peppermint was interrupted from her musings by a gentle, silvery voice. "Welcome dear! Why don't you introduce yourself?"
Peppermint turned her head and realised that it was the princess who spoke to her.
She wore a smile on her face and answered. "Your majesty, my name is Peppermint Icicle Frost."
Princess Celestia stared at her and murmured softly, "Peppermint... what a beautiful name." Then, she said aloud, " Why don't you take a seat next to Twilight here?" Gesturing to a lavender filly with her nose buried in a book.
Peppermint made her way to the small table, and sat on the chair next to Twilight. The unicorn smiled at her but went back to reading her book. Peppermint heard the voice of her teacher filling the room and sighed happily.
This school was one she had been yearning to attend since she was little. Her father used to urge her to learn proper magic, since he was a magically skilled unicorn himself. So she started reading any spellbook she could get her hooves on, and also historical books on magical events and sorcerers. In one of them, she read about Celestia's School for Gifted Unicorns, and that it was the most wide-known magic school in Canterlot for a thousand years. So she practiced and worked hard to pass the entrance examination to enter.
When the day came, the inside of her stomach was swimming with snakes. It was a tough spell, to hatch a phoenix's egg. Every ounce of her magic energy was concentrated on the egg, but it refused to budge. Then, she remembered something she had read in a book about biology. Dragons blow fire on their eggs to warm them up and make them hatch faster. It might work on phoenix eggs too. Peppermint exhaustingly casted a heat spell, and surprisingly, a beak popped out of the egg. She started crying and laughing at the same time.
Peppermint had made new friends at the end of the day. Twilight was really helpful when it came to making potions, and was a really talented filly in Peppermint's opinion. Lemon Hearts and the others showed her around the school, and what fascinated her most was the huge library with its endless bookshelves stretching out on the hall ways. Combat duels were her favourite, and she showed a great interest in them. Princess Celestia smiled knowingly when Peppermint eagerly fired a spell at a training dummy. She also paid attention to Princess Celestia's lessons about legends.
In one of the lessons, Princess Celestia explained on how an empire in the far north was cursed to disappear for a thousand years by an evil king who forced his subjects to become slaves. She lead an army to fight the king and his bewitched soldiers, but before the king was defeated, he placed his curse upon the empire and it vanished.
When school was over, Peppermint slowly trotted towards the gates of the school, when an elderly guard walked from the direction of the Canterlot castle and came over to her. When he saw her, he smiled and walked over to her.
She slowed down and uttered in a confused tone, "Um... can I help you, sir?"
"I'm sorry miss, I just wanted to ask if you belong to the Frost family."
"Yeah... I'm Gleaming Frost's daughter."
"I can surely see the resemblance. Your father and I are friends since he served the Royal Guard Service. My name is Storm Shield."
Peppermint was about to answer when she saw Lily Mint landing on the stone path outside. So she said, "I hope to meet you again, Sir Shield." And she waved goodbye to him and trotted over to her mother.
Lily Mint was panting as she nuzzled her daughter. "How was school dear?" She asked.
She replied, "It was simply magical, mother."
Between the desk? Um... I'll try to do a visual of that: "Legend: T is a column, L is part of the desk." LTTL See the problem there? The columns are quite crushing. (Had to get that out of the way, great opening paragraph! The picture is distracting, but nothing too bad.)
Alright, I'd say this chapter was better than the last. Still some problems, but it did much better. One of my main concerns just so happens to be the formatting you are using. At two different points you combined paragraphs or spaced out something that shouldn't have been. The first example being the exchange between Peppermint and Celestia, the second sentence... er, third sentence is moved to the next line; with the line from Peppermint being directly below it rather makes it seem like one whole paragraph.
Be sure to properly space things. The other part was right at the end. Lily Mint (Mint) arrived to take Peppermint home, before asking how her day was... to which the answer comes -- but instead of being a new paragraph it is dropped down right below. You could have even done into Mint asking instead of just mentioning it. A good rule of "thumb" (fetlock for the pony version) would be this: "Show don't tell." It doesn't work in every situation, but it is just something to keep in mind. Summarizing events is an example of a "tell time"; while a conversation between two characters is usually a "show time" (lol -- I did not mean to make it say that).
There were also a few points you could have expanded, I'll grab them and list 'em: "This school was one she had been yearning to attend since she was little." Why was she yearning to attend since she was little? That could have a paragraph or two by itself, or maybe a spot later in the story. "Her teacher however, didn't seem too happy about her reading that one." How was it Peppermint noticed? Expand that some, as it is strange to just note that someone doesn't like something; perhaps mention that the teacher gave a questioning look, or something along that line. An explanation, that's it! "When her mother Lily Mint came to pick Peppermint up, she asked her how school was." How did Mint arrive, was she on-time or late, and what happened when she got there? As it is we know she arrived, and asked Peppermint a question. But that is it.
An error would be a missing comma. I'll grab and mark the sentence, blue for the fix: "Her teacher, however, didn't seem too happy about her reading that one." I'd honestly say that it would flow better with adding the word 'all' right before 'too'; that is just my thought. I didn't notice anything else... maybe I missed them? Maybe there aren't any others? I don't know.
Back to the picture. Lyra! I found you~ Er... right! While the picture does help to paint the picture (heh) it helps to not rely on it to show the readers the scene. It will be so much more worthwhile to be able to describe the scene and let the rest fall into the readers imaginations. The extra room is something that really helps to get a reader invested (at least for me, I'll have to ask others to see if it is the same for them). I will also add that sometimes a picture is a fun thing to see, however, it falls into the same thing. Let me just put it a different way: The picture should help show the scene, not define it.
Finally, I wanted to address character: Celestia is easy going (especially for a Princess) yet she seems a little too lax in this chapter, and she is motherly. I mention the latter because she would be benevolent as she teaches classes, yet something I have noticed is that she keeps to a certain formality; or rather she doesn't stop others from doing so unless they are on the same status (i.e. Twilight once she became a princess) -- otherwise I don't see her telling somepony to be less formal. The opposite can be said as well: if the pony starts without formalities... would she ask them to address her with formalities? I do not know.
I'll be seeing you in the next chapter... and silly me forgot to mention. Feel free to ask questions, or reach out for help if you ever need me. I'll respond as soon as I notice that there is a message (and the time it takes to write it). Anyways, I'll be off for now. Almost forgot, again. You're welcome, it was my pleasure to be of assistance.