Yamamoto's first seen in canon as an apparently normal student, if one that's fairly popular and well known; decent grades, but obsessive about his place on the baseball team. After overdoing something in practice and breaking his arm on what was possibly a misfire of following advice from Tsuna, he's left facing a decent chunk of time being unable to play at all... and this is where Tsuna ends up saving him from a suicide attempt.
You will pardon MUN's doubts about his mental health at the beginning of canon compared to the happy-go-lucky image he very obsessively projects.
Now feeling somewhat indebted to Tsuna, and attached as well in that there hadn't really been anyone else in the school willing to put themselves at risk to save him, he gets pulled into being part of Tsuna's "Family" by Reborn - Tsuna's mentor, training Tsuna and the others as successors to a very eccentric group of Mafia and underworld.
(The larger history of the Vongola Family is loopy and they sort of became such by accident, but that's a story for another time and their founders; they don't really function that much like a normal Mafia as a result. In 30 Seconds With Bunnies form, the founders were a group of revolutionaries in the Italian Unification war that started out FIGHTING the early mafia, and anyone else who would abuse power or pick on the weak; as things settled with them conflicting less with legal authorities, they basically were only working in underworld and criminal circles trying to impose some kind of limits and morality boundaries where legal authorities couldn't reach, which led to them existing as a criminal entity themselves. One of the off-rails founders tried to make it a more vicious form of such, but they're still very weird and more weighted towards the original intent. Telegraphed pretty loudly in canon clues, Yamamoto's great-something-grandfather was one of the founders; the family line still served the intent even if they were separated from the main organization. His father's only half-retired and is on a first-name basis with Reborn and with Tsuna's father who is actively involved; Yamamoto approaches his father for training in the family's swordsmanship and does a few things acknowledging awareness of the family history early on.)
Reborn never directly threatens his bubble of denial, but does start casually giving him weapons and putting him through tests and training bits, noting out of his earshot at one point that he has the reflexes and instincts of a natural assassin. Yamamoto also demonstrates an amazing talent to basically wander into things, or show signs that there's a little more going on than what it looks like, including a chapter or so where he's absent for part of it because of ending up on a trip to Italy with the Boss of an allied Family after carrying a message for Gokudera. Despite some of Reborn's training exercises involving live fire and said incident, and a squabble with the local Yakuza, Yamamoto cements an early habit of playing completely oblivious and referring to what's going on as "Oh, you mean the game?", after an early moment of lampshading Reborn's recruitment as "Aww how cute the little kid's playing mafia!".
(*For the canon-blind, Reborn spends the series suffering from a curse that keeps his physical body that of a small child, even though he retains all of his mental faculties and limited use of his powers.)
Considering that, even early on, he manages to not slip up in a way that'd be dangerous to him or others in spite of his insistence that it's "a game", he caught on fast and had a good clue what was going on and just refused to admit it.
The relatively less serious juggle of chaos ends up getting an abrupt shift with a series of attacks on Namimori students, a set of attacks that slowly zeroed in on Tsuna and his group - the work of a group of fugitives from an underworld prison that were attempting to hunt down the Vongola's successor, Tsuna, for an unknown purpose. Tsuna's charged with tracking down and recapturing Mukuro and his group, leading to an expedition into the abandoned Kokuyo amusement park where the fugitives are lairing. For Yamamoto's part, he's at first cautious to the point of crippling his ability to fight, until Tsuna's shoved into the path of the one after Yamamoto - at which point he not only draws attention back to himself, but allows an arm injury as a way to get close enough to disable the enemy, something that did potentially put him at risk of being unable to join the baseball team for the end of season games. (Remember that shortly after crazy things started happening around their school because of Tsuna and Reborn, a broken arm preventing him from playing was a last-straw on his suicide attempt; taking an injury that puts him out of the game is NOT a light thing for him.) He ends up getting battered around a bit more not long after against what, at the time, they'd thought WAS Mukuro, enough to be put out of commission for the rest of the arc, albeit after figuring out a few key clues on the enemy at the time.
Tsuna catching Mukuro had one small, problematic side effect - it brought their group to the attention of everyone else, and led to a challenge for the succession, something announced by the Varia's Rain Guardian, Squalo, pretty well thrashing their group and brushing Yamamoto and Gokudera both aside. While there was a brief point during the challenges where Yamamoto'd started to question if they shouldn't back out and keep themselves intact as a priority, there'd been enough of a blow to his pride for having to give up that the idea of a chance to even the score with Squalo did get him switching from "Maybe we shouldn't..." to "Oh hey this could be worth it :3 " when it came down to the challenge being resolved via a series of duels. Even with that demonstration that he does have some pride that can get bruised and can turn viciously competitive, he put himself at risk again trying to make sure Squalo survived the incredibly hazardous setup they'd been put into; he was shaken badly when it looked like he'd failed to save his opponent - and visibly relieved on finding out later that Squalo had survived. He also manages to restrain Hibari, the most dangerous and volatile of their group, briefly and with almost no fear when everyone else is getting out of the way, and continues to backup, support, and go straight to cover for whoever's weakest as the situation deteriorates into chaos.
And yes, he goes through this entire thing, with all of the moments of acknowledging life or death stakes and everything, with occasional periodic moments of lapsing into his idiot routine and "Oh, the game!", just to reaffirm his denial bubble to everyone else.
With enough time for things to settle, the group gets gradually thrown forward thanks to a misfire of a piece of technology to a ten-years-later future; Tsuna's the first one thrown forward, and gets greeted by Yamamoto's older self. This gets notable for the fact that older!Yamamoto doesn't even ATTEMPT to play denial, discussing Vongola business and what's been happening so far naturally, and starts trying to help the younger ones catch up on what's going on...until the swap happens and his younger self gets brought forward in his place. He spends the entire life or death struggle to stave off that future going dystopia working on his sword training and being perfectly competent, to the point of having a rare fight with Gokudera about the way Gokudera's been hostile to the rest of the team despite talking up being the "right hand and second in command"...
And even while he's fighting that seriously, the argument with Gokudera is one of the few times he drops the act of pretending he doesn't know what's going on.
The denial bubble gets more transparently fake with the note that Reborn gives him a challenge with "I'll tell you my past and everything" as the prize, and Yamamoto passes the challenge; while whatever Reborn shares with him stays off camera, he's not attempting denial-bubble behavior when he's alone with Reborn leading up to that and afterwards, and it's not a conversation that would work at ALL if Yamamoto had been trying to insist "it's a game".
Still, yes, he goes through basically a small guerrilla war still trying to pretend it's a game and something that isn't his first priority in life as an outward face, despite acting with the same caution and regard for his teammates, and willingness to self-sacrifice, that he's always had. Somewhere late in it, he does make an admission about realizing that he can't keep trying to cling on to other things and still protect the people he cares about, but he never outright admits that he's accepted and aware of what's going on...
And on the eve of their return to the past, he expresses being glad to go back to baseball and not worrying about this sort of thing. "I never said I'd keep focusing on this when I went back." At the same time, he's doing the same sort of thing he's prone to where his side of the conversation is cheerful and mostly normal while the other party can be doing anything up to and including death threats in their side without him blinking at all.
Not long after, while they were still readjusting, two things started up. One was preparations for Tsuna's official inheritance as the heir of the Vongola, an event that had drawn a lot of attention, some allied visitors there to pay respects and come for the ceremony, and a good number of assassins from less allied groups wanting to pick off the heir; the other was a small unknown group showing up for the ceremony that signed up for classes at their school and was quickly working to get close with their group. While many of the Simon were quick to strike up friendships, it was a randomly distant thing; Yamamoto happily accepted the one joining the baseball team and hanging around him, but had the brilliant idea to start fishing at having realized they were Up To Something while alone with his back turned, and got run through, barely managing to leave a short warning in blood for the others on a scrap of paper.
For most of the rest of the arc, while the others are rescuing a kidnapped member of the group and fighting to figure out what the Hell's going on, Yamamoto is in a hospital with doubts he'll ever walk again.
Late in it, the present-day, their-timeline version of the man that'd been responsible for EVERYTHING in the future arc - aka "his alternate timeline older self is one of the few people Yamamoto hates" - shows up in the hospital with something to heal the damage; while Yamamoto's not fully recovered by it, he's at least up enough to play backup, and get the drop on the body-hopping long-dead illusionist that's pulling strings briefly. Not enough to actually STOP him, but enough to startle him and save a couple people he was about to kill.