"Cat, we're stepping into our future and you're not even paying attention."
Catrine’s hand halted, palm pressed against the cool surface of the heavy wood door. She raised her eyebrows at David and made no further motion to exit the main building of the Academy.
Green highlights, throughout: vagueness and/or weak verbs. These are things that would read better if they were more specific and/or more active.
"Our future doesn't start for another couple of weeks," she said.
They were allowed to leave the school that day because the council had granted them an audience. Graduation was still two weeks away.
"You're wrong. This is it. Today we make history." David pushed the door open and the mid-afternoon light rushed into the hall.
She chuckled. "We should wait until the council approves the Tutor Program before we celebrate."
"Why would they reject it? You wrote a great proposal. The argumentation is flawless."
"Just because you couldn't find any faults, doesn't mean the council won't."
Show vs. tell in dialogue -- this dialogue is okay, but it doesn't sound like real people talking. Cat and David both know why they're going to this meeting, what was in the proposal, and when graduation is. Why would they talk about it, then, unless there were lingering questions or problems? They're only saying these things so the reader knows them.
How often do you turn to a friend and say "Today I make history," -- and not mean it as a joke? (it can be a good joke, admittedly)
This kind of narrative dialogue turns up a lot in TV shows. CSI is especially guilty, since they have to explain procedures and logical connections to the viewers. But real people do not talk like that.
May I suggest explaining the proposal in the narrative -- and keep it very short, because you can explain more once we get to the meeting. Plus, it's not actually that important. Lead with drama: Cat's fears about the proposal, graduation, her attraction to David, or whatever's appropriate. Another problem with narrative dialogue is that it's low on emotions. Low emotions = "slow start".
They walked toward the front gates of the Academy. The wooden bars stood wide open and seemed more decorative than a true barrier to their exit. Catrine’s eyes were still adjusting to the brightness, but they took in the novelty of her surroundings. This was the first time either student had left the school since their enrollment at the age of two. They were both eighteen now, but Catrine did not feel as ready to conquer the galaxy as David was. Her insides were twisted in knots. She had eaten hardly anything all day. But she would not let her nervousness show--not even to her best friend.
Yellow highlight: this is the only "tell" in the narration that you might want to "show" instead.
"You're fretting too much," David said. "Everyone in the department loved our idea to teach the rest of the galaxy and the council will too."
After this, the dialogue gets considerably better.
Deep down, Catrine knew they were as ready as they could ever be. She had spent weeks writing the proposal and they had prepared David's speech with great care. But it was such an ambitious project...
"I'm just glad you will be the one doing all the talking," she said.
Their dark gray tunics and slacks sparkled in the afternoon light. The uniform designated them as students from the Governance Department, but no one prevented them from exiting the Academy grounds even though they were unsupervised. David had managed to convince their instructors there was no need for a chaperone. He thought the lack of one would increase their chances of successfully defending the proposal. Catrine had not argued with him, but she had made sure she had clear instructions on how to get to the government building. The last thing they needed was to get lost on the way.
"Wow! I've never seen so many flowers in one place," David said as he stepped through the archway and out of the Academy grounds.
Catrine followed him toward the flowers. The bright colors on the other side of the entranceway grabbed her attention like David's raised voice in the quiet reading room. She inhaled deeply. The floral tones drifting from the garden gave the air a sweeter scent.
"Now we know why everyone talks so much about the Center Gardens," she said. "The directions say to go around it."
"We have time. Let's walk through the park."
"We don't want to be late..."
"Then don't waste time arguing." David grabbed her hand and led her down the footpath closest to them.
She knew better than to argue with him--especially considering she wanted to stroll among the flowers just as much. Catrine noticed flowers of different colors planted in sequence and she wondered if the colors represented the diverse departments of the Academy. The path in front of them started by a simple two-tiered stone pool and bowl adorned solely with the letter W three times around its circumference and led toward the larger fountain they could see farther ahead. The violet beds on either side were in different shades of blue, dark-colored at first then lighter the deeper into the park they walked. The fragrance of the flowers, on the other hand, intensified toward the middle of the Center Gardens.
Did you mean to hint at David being violent?
The perfume in the air stirred Catrine's empty stomach and she soon regretted their route. She squeezed David's hand for support when her step faltered and he halted.
"What's wrong?" he asked. "You look pale."
"I should have eaten more for lunch."
"Let me help you to the fountain. You can splash some cool water on your face."
He slipped his arm around her waist and her heart rate doubled its pace. Catrine took a deep breath to clear her head, but the cloying smell only made her dizzier. She staggered the next few steps towards the lilies that surrounded the fountain, leaning on David for support. She needed to get away from the perfume of the flowers, so she buried her face in his chest to mask the scent.
He enveloped her in a hug, his hand gently stroking her back. "Relax, Cat. You're just stressed about the proposal defense. Everything will be fine."
His voice was low and soothing. When she looked up, David was smiling at her. She wanted to thank him for being there, so she rose to her toes to give him a kiss. But instead of the friendly peck on the cheek they had often exchanged, their mouths joined in an embrace that did not let go.
Catrine tasted the sweetness of the lilies on his soft lips. It mingled with the alluring smells around her and the enchanting melody of trickling water coming from the fountains in the park. The kiss that had started as endearing gathered intensity like the scent of the flowers throughout their stroll in the garden. Catrine interlaced her fingers with the hair at the nape of David's neck and pulled him even closer. His hand reached under her tunic and seemed to burn the skin it touched.
Then David broke off the kiss gasping for breath and stepped back. "We don't have time for this. We're going to be late."
Neither of them seem particularly disconcerted by this passing flash of lust... so I have to assume this has been going on for a while? If it hasn't, you're passing up a gold mine of conflict and reader hooks.
Catrine had no time to process what had happened. David grabbed her hand and dragged her through the violet bed, heading northeast. The sea of flowers they trampled changed from blue to purple to magenta. When they reached the path between the red
Highlight: kinda awkward wording, there. Past tense of shine is shone.
David had not looked at Catrine during his rush to leave the garden, but he had not let go of her either. When they reached the front door, he gave her hand one last squeeze before releasing it.
"Relax," he said.
Catrine was still trying to catch her breath, but she noticed David's back straighten and she mimicked his posture. Then she entered the main government building a couple of steps behind him, at a much slower pace than her heart rate.
To answer your questions directly: yes, it's a kinda slow start because you're not tapping the emotions I suspect are here. Your show vs. tell ratio is fine, in the narrative. I already talked about the dialogue. On the whole, I think all you need is to shift the focus a little bit.