The shinigami academy is a place for all prospective shinigami to learn all about their future profession. Entrance into the academy requires a passing grade on a written exam, and the presence of reiatsu. Anyone from any class, district, or age can apply. When accepted, they are each given an asauchi, or a nameless sword, which they must tirelessly imprint with in order to awaken their zanpakutou. The students learn how to control and manipulate their reiatsu, communicate with their zanpakutou, and utilize the four main forms of combat over the course of six years. Hitsugaya did it all in one.
Unfortunately, we do not get to see Hitsugaya during his academy days in the manga. We know that he graduated within one year, and is the youngest graduate to date, having broken Gin Ichimaru's record. It can be assumed that he woke Hyourinmaru early in his enrollment, as he was already seeing and hearing him in his dreams long before he got his hands on an asauchi. He graduated at the top of his class, with the highest scores in both written and performance exams, and was recruited into the Gotei 13 directly from the academy.
He is considered a prodigy and genius by those around him, called so by Abarai Renji and Iba Tetsuzaemon. Gin Ichimaru refers to him as "the embodiment of a heavenly guardian that is reincarnated every few centuries." Even Kyouraku Shunsui, one of the most powerful shinigami in all of Seireitei, states that Hitsugaya has the potential to surpass him within one-hundred years, which we can assume is not as long as it sounds when shinigami can live for centuries.
It isn't long at all before Hitsugaya is a seated officer in the Gotei 13's Tenth Division, under the leadership of none other than Shiba Isshin. Hitsugaya becomes the third seat, ranking below only the Captain and the Lieutenant, a place held by Matsumoto Rangiku. Unfortunately, these two are both not very reliable when it comes to the mundane tasks of Division leadership, and are more likely to be found goofing off rather than working.
❝What took you guys so long...? I've finished filing the reports.❞
Hitsugaya showcases his ability to pick up the slack early on, finishing the paperwork of both the Captain and Lieutenant while they were fighting over actually doing it. Isshin does everything he can to avoid doing it, and Matsumoto does everything she can to make Isshin do all of it. Which means it is left in the hands of the always-capable yet ever-burdened third seat. Because someone has to do it, so let's leave it in the hands of the shinigami equivalent of a ten-year-old, who is clearly more capable than the adults around him. He seems doomed to be forever exasperated by the adults around him.
Isshin is overjoyed by Hitsugaya's willingness to work and ability to compensate for his own flaws, lifting him in the air like a proud papa. Hitsugaya's total lack of reaction means that we can assume this happens often. He doesn't berate him for treating him like a child, or for slacking in his work, and just quietly accepts his fate with the resignation of someone who has known true suffering.
❝At this rate, you'll be the next Captain!❞
Isshin happily remarks that Hitsugaya is on track to take over the Division, quickly sparking a small argument with Matsumoto over their rankings, and how she is next in line for the position of Captain. Isshin thinks the Division would fall apart under her leadership. This sounds like very heavy subject matter, but this whole part of the scene is lighthearted and fun.
Hitsugaya (still suspended in the air and totally deadpan) is quick to agree with the Captain's assessment.
❝My bankai training is almost complete. It's only a matter of time.❞
Matsumoto scolds him for not having her back, and it is really just fun to see the dynamic between the two of them when she outranks him. I adore seeing lighter moments involving Hitsugaya because they are few and far between, and it is sometimes easy to forget that he isn't always such a serious character. Even if he pretty much never stops glaring. Isshin is much smarter than he leads people to think, so I believe that he knew exactly how strong Hitsugaya was and was serious when he thought he would become the next Captain. And Hitsugaya did take over after Isshin no longer held the position. The fact that Matsumoto never says a single thing about it in complaint and holds no resentment toward him whatsoever over skipping a rank indicates that she cares very little about becoming a Captain. But she'll still pretend she does in order to tease Hitsugaya.
Later on, Hitsugaya informs Isshin of a concerning situation regarding disappearing shinigami. This matter is what ultimately leads to Isshin's exile and disappearance from the Gotei 13 and eventually, the birth of Kurosaki Ichigo. Alarmed by the situation, Isshin rushes off (without informing the Gotei 13, naturally) to see the situation for himself. Matsumoto tries to follow after him, but is stopped by Hitsugaya.
❝As you and I are now... we'd only be a hindrance to him.❞
No matter how much power someone has, how quickly they rise in rank, or how much of a genius they are, they still have limitations. Hitsugaya acknowledge his here, and though he is concerned for his Captain's safety, he has faith in him too. He would rather stay behind and wait for Isshin's return rather than get in his way and hold him back. He showed this inclination for putting the well-being of those he cares for above his own many times before this. There was the time when he made sure to send Hinamori to the academy with nothing other than their normal bickering, so that she felt comfortable leaving him behind. He also didn't want to leave his grandmother alone.
Despite the things he says, Hitsugaya is always thinking about his impact on the ones important to him. As much as he would love to fight beside and back up his Captain, he simply feels he isn't skilled enough yet to do so. This shows that his earlier proclamation of taking over the Captain's position was nothing but empty, false arrogance. He knows he is powerful, but he isn't conceited enough to believe that there is no one stronger. In contrast, he never once considers his subordinates to be people who get in his way or hold him back, even if they are weaker and he has to save them or is injured because of them. He only holds himself to such idealogies, and that says a lot about his character.
When Isshin leaves a second time, this time with no notice except for a handwritten letter, Matsumoto is furious about his unannounced disappearance. At that point, there is no hint of what is to come, and she simply rants about him abandoning his duties. Hitsugaya is not amused by her hypocritical outburst. This scene is meant to be funny, but it is the last they ever hear from their Captain. We can assume that it is shortly after this that Hitugaya received his promotion. Despite believing himself to be too weak to help his Captain, he is still more than strong enough to take on the position, and is the youngest Captain in Seireitei history. He was already doing his Captain's work before he took over the Division and is an accomplished shinigami who has achieved bankai (another record for the youngest in history.) He is more than qualified to take the new position and does so with pride. And maybe a hint of annoyance.