If I were to use training, then I would have characters require training at ASI levels; you train in an ability score(s) or a technique (feat). During this time, your class trainer will teach you concepts and ideas that your character will piece together throughout their experience adventuring. They would gain HD, proficiency bonus, learn new spells (as they gain a better understanding of the manipulation of magical energies), and new class abilities (as the metaphors and concepts their instructors taught begin to piece together in the character's mind). I feel that the ASI levels, varying wildly from class abilities (a fighter could take magic initiate druid, and their regular instructor wouldn't really have any good reason to be able to teach them that, barring special circumstances), are the most appropriate levels to require training for, aside from the subclass levels.
For a planned multi-class build, I would have the character still have an instructor who utilizes a similar build, and just make a name up for the build and call that their class. For an un-planned multi-class choice, I'd only grant them the hit die, proficiency bonus, and any abilities they could learn from another party-member (divide casting into arcane, divine, primal, and one like-type member could give basic instruction in the spellcasting, but restrict it to cantrips if the class gives cantrips, then there are classes that grant skill proficiencies, if a party member has the skill they want the other party member could teach it), and then make them search out a teacher and convince them (persuade, deceive, bribe, threaten, whatever) to instruct them, and then follow the above "at ASI level seek me out again" guideline. Feats you seek out specialist teachers for, pay them, and then they teach you the skill. I don't know what kind of timeline I see being appropriate. I feel like a month, flat, across all levels is appropriate, and during that month, you have to make time to spend two-weeks learning a feat as well, so you can only really rabble-rouse for a solid week when you get home, start training your main skill during your afternoons, then you start to take it seriously and start training during the day, and then for two-weeks your evenings are spent training a feat or ASI, and then to celebrate and get some partying in before you head back out, you start training in the afternoons and partying during the nights for the last week before you head out on another adventure. Gives you roughly 30 days to put towards a skill or tool proficiency as well if you don't spend all your free time carousing.
Tangent:
I have never run it the above way. I did run one game where it was the general consensus "on the job" style, that game was a side-scroller and I ran it like a cooperative Metroidvania game; you level up at the end of a fight or when you get XP sufficient to gain a level, you get full HP, and if you would get new slots, you get the slots you gain. I just wanted to see what kind of problems (aside from platforming with the way jumping works) a side-scroller design might pose, and without flat-out banning flying type effects, and radically changing jump mechanics (basically, no roll jumping, you jump x squares based on str or dex, far or tall it's squares equal to modiifer, but then you can actually do some pretty fun platforming, and archers are the straight up bomb to use as enemies when you can include so much more verticality in your encounters and limit the path options and maneuverability in general), I don't think it's particularly great. At low levels it's fine, you can have some really fun encounters, but once you start to get up in level, especially with the video game style buy whatever you want if you got the gold setting, things escalate quickly. Started them out at level 3, by 6 all but one person flew, and I gave him a hookshot type item because he was a cleric and had low str and dex and couldn't really platform very well. He rolled a lot of athletics checks to climb since he could just reach ledges with his strength. Flanking isn't as easy to set up either since melee tend to group up, which makes it less powerful to allow. Players start looking into ranged options as well really quick once they've been in a combat or two and realize that they're not always going to be able to get in to melee and get off the ol' stabbity stab.