I like it so far. You started with a conflict, which is sweet, and people are going to want to continue reading it because they want to know just what Allen and Roy are talking about. I think, writing-wise, you just need to watch your unnecessary words (which takes a lot of practice).
I went through the whole thing and put in some comments/suggestions in red. I didn't actually remove anything (that's your job

) so feel free to ignore or do different!
Part One
The young adult
[Perhaps “the young man?” “young adult” could infer either male or female] gave a deep sigh [perhaps “sighed deeply?”], disappointed in how the conversation was progressing. He scratched his short brown hair and thought of some way to approach the issue better, though with no real success.
“Hey, can’t you at least try to make things a little less difficult?” he said, desperate to
break through to the other person on the telephone.
“Why should I care? It’s not my problem. It’s not essential,” the male voice on the other
end of the phone replied.
“Think a little bit more about others, man. It’s not exactly easy dedicating myself to you
and crap. Catch my drift, Allen?”
“I’ll ‘catch’
[‘Catch’ does not need apostrophes/quotes in this instance] you in class, Roy. Goodbye.”
[Perhaps just a plain “bye?” Sounds more realistic. Or “Good-bye” if you want to put more emphasis]
Roy stared blankly at his phone as it flashed the text
[“the text” is unnecessary and can be deleted] “Call ended”. After another routine
sigh and a glance at the clock he laid back on his bed trying to remember how he ever became friends with Allen in the first place.
Part Two
It was odd being a few years older than most people in the classes at Veridian
University, but Roy at least thanked the stars
[“thanked the stars” is a bit cliché, don’tcha think?] he had someone just as odd to hang out with
—the creepy, white-haired Allen, his fuzzy hair cut short and simple to match his always-clean always-professional look. It was sort of a wonder how the two were friends—one, a former boxer who volunteered in the military for four years on a special contract, and the other, a seemingly genius student who never forgot a name or ancient date in history.
Indeed, they were an odd pair at first glance, but Roy had actually always wanted to be an engineer, not a boxer. However, the war between the Northern Alliance and the Southern Federation made it difficult for people of his generation to even make it past the age of 25, so by the suggestion of his parents, he volunteered in the military and came back alive—an impressive feat that many attempted and few succeeded at.
Smart as he was, Allen was no weakling either though. His body wasn’t nearly as built or sturdy-looking as Roy’s, but he could hold his own in an arm wrestle against Roy, a fact that no one believed until Allen was convinced to partake in a series of arm wrestling challenges that finally validated Roy’s seemingly preposterous claims.
He had no visible fears or weaknesses either—most everyone gets a little ticked off or frightened sometimes, but Allen only had ever
had one expression on his face, as if though he were oblivious to the existence of other ones.
[Holy crap there are a lot of dashes. I’m a big fan of them myself, but definitely try to reword some of these sentences so you don’t need to use so many. And by dashes I mean the long dashes between phrases (–) not the short guys.]
As the two walked through the hallway to their Physics class, they heard the sound of
the TV in the nearby lounge, as loud and disruptive as ever. Today it was turned on to the news, like any other day
[if it’s “like any other day” you don’t need the word “Today” at the beginning of the sentence. You can start with “it” and achieve the same thing] , once again covering the story that people had heard and grown tired of already where twenty-six students were shot, only six of whom survived
[alternatively you could say “and only six survived.” I only say this because I generally dislike the word “whom”
.] There was no new news to report, unless one considered the lack of news in itself noteworthy
[Perhaps “newsworthy” for extra pun? No? Okay, fine…]. The lounge and hallways in general
[“in general” is unnecessary here—delete without fear!] were empty, suggesting Allen and Roy were either a few paces behind or
ahead; in this case, it was the latter, as the Physics room looked so empty one might have thought everyone had gotten lost on their way to class
[Maybe instead of being late you could say “skipping class because the first day is always syllabus day?” Might be funnier].
A few students and their buddies murmured in the chairs of the large lecture hall—
the class had over couple hundred students as most people in their second year took the class.
[Snow pointed out this error, I think]. Allen and Roy decided to sit up
[“up” is unnecessary here] only a few rows up
[also here] from the front, despite having the choice to sit almost anywhere they wanted. It was just a choice on whim
[kill this sentence—unnecessary filler].
In the meantime, the Physics professor spared no time before beginning his lengthy
introduction to the class. As soon as the seconds hand on the clock ticked to mark the, well, technical commencement of the class, his voice echoed through the mic so that everyone who didn’t want to hear him not only could hear him, but would hear him.
“Welcome to Physics 122 on this beautiful afternoon at Veridian University, located
between... and... in the city of Alpha, in the Northern Alliance... continent of Earth...” he went on, giving such an elaborate description of the class that it sounded more like he was telling the setting for a story.
As the people already situated in the room tuned out his familiar voice,
other voices began to crowd the room through the doors at the back of the lecture hall,
followed by the footsteps marching down the slanted ramp down through the rows of chairs. It
seemed the Professor was used
to this as he paid it no mind and continued his introduction—
perhaps it was his way of making a few “free” bucks at the beginning, or warming up his voice for the rest of the hour lecture. People usually stopped coming up with theories after the first couple of weeks passed by.