By making the training itself the Hero's Journey
If you haven't already, I strongly recommend picking up a copy of Christopher Vogler's The Writer's Journey. While it is, quintessentially, a screenwriting text book, it is still extremely valid and a great resource for writing in general.
In short, it promotes a refinement of Joseph Campbells The Hero's Journey, and breaks the Three Act Structure into 12 smaller stages. It also further explores some major character Archetypes that are relevant to your story.
Essentially, it gives structure to your story and promotes the growth of your characters through their individual journey. It's adaptable to most stories, and isn't reliant on external conflict, and it rewards the reader with a fulfilling narrative. It's a timeless structure, and it works.
So how, then, can it be applied to this particular case? At it's core, the Three Act Structure and the Hero's Journey (particular Voglers) revolves around:
Act 1: Departure
This is where our story starts, we are introduced to the protagonist (our Hero) and the "ordinary" world. We hear the call to Adventure, meet our "Mentor" and take the first step/pass the First Threshold.
The call to adventure can be when our lowly Hero is selected for this particular training course, either willingly or unwillingly. Maybe his initial refusal causes some personal catastrophe for himself or someone close? Maybe he's seen something or survived something and wants to make a difference. Whatever that inciting incident is, this is where it sits. What pushes him through the training is defined here.
Act 2: Initiation
This is where the training really kicks off. There are tests, trials, failures and lessons. As the Hero's skill increases, so to do the challenges and risks. Then comes the big challenge, the approach to the "Inmost Cave", the Ordeal and the Reward. This is the final struggle the character has in the training arc, often with themselves. This is where they overcome their own fears and doubts, push themselves beyond what they thought capable and emerge triumphant. Think along the lines of Luke on Dagobah and proceeding on to Bespin.
Apotheosis occurs, the Hero overcomes the final trials and is made stronger for it.
Act 3: The Return
This is the end game. Using their new-found skills and abilities the Hero returns to the ordinary world or continues on to their ultimate destination. Their training complete, they are pushed back into the real word and ready to face the challenges. There's still struggle, still trials, but the Hero is equipped to deal with them and has the support of allies, who will ultimately play a role in calling the hero back.
In the end, it's important to remember that it's all figurative. Conflict does not have to be external or violent. Antagonist do not have to oppose the protagonist in such a manner that the only way to win is through outright violent conduct/conflict. The Hero's Journey is about overcoming obstacles and trials, and growing. It also doesn't have to be the be-all and end-all of your story. Incorporate it into your existing work if you aren't already following it. Use it in miniature to express the training and development of your character. If you want the training to be a focus, then make it the focus.