Another unusual situation that is not the norm.
Player characters aren't the norm. They are exceptional, unique, special. It's even canonical in an official D&D setting that you could become a Cleric by divine guidance, Goldmoon from Dragonlance (the default Cleric PC in the original modules) became a Cleric by pretty much having a vision and suddenly finding out she could pray for spells and recieve them, and just by reading the Disks of Mishakal (that were effectively holy scriptures engraved on metal discs) characters could become clerics after a period of reading the discs and meditating on them. A cleric is just a character who prays to a god and the god answers directly, everything else is a setting presumption.
Some kind? Well martial arts certainly requires some intense training and if mages in the game have to be half as well informed as earthly members of occult orders ie. The Golden Dawn, Thelema, Aurum Solis, etc. (read some studies on esoteric Qabala sometime to see what esoteric training would be like) then they are certainly apprentices for years before becoming real wizards.
This is D&D, not anything "real", practitioners of real-world occult traditions spending years to master the teachings of their order has
absolutely squat to do with D&D. If you want to make becoming a 1st level wizard take many years of study, go right ahead, but that is another setting-specific consideration, not innate to the game. To others having a wizard show them the basics for a week or two and many hours of practice could suffice (at least "well enough").
1st level Wizard doesn't mean they are deeply versed in the arcane and occult traditions of their world, it means they know the most basic principles of the class, how to begin learning on their own (with XP) and perform the most basic tasks of their class (like casting spells). It isn't even a third edition conceit that you can learn to be a D&D wizard that quickly. The old AD&D "Treasure Hunt" module (TSR 9185) even includes an explanation of this: the module is about normal people who survive a shipwreck and gain their 1st PC class level surviving on an island, and you can become a Wizard by spending several hours reading and experimenting with some spellbooks and wizardly acoutriments that survived the crash. Not a master certainly, but enough to pull off a Magic Missile and call for a Familiar (i.e. 1st level).
IMO the idea that classes can be picked up with a minimum to no training at all must mean that in these settings there are no apprentice mages, acolyte clerics, novice martial artists who are older than adolecents or perhaps only the inept need to apprentice at anything at all.
A young student who wants to learn to be a monk could spend 5 or 10 years in a monastery learning the intricacies of philosophy and many days in simple labor punctuated with lessons in martial arts and spiritual training, or an old master could find a talented new student on the eve of a great war and give him a crash course in the fundamentals of their fighting style and doctrine, which he gets to put to lots of practical use in battle and become a 1st level monk when he levels up. Condensing loads of training into a short cinematic period where they learn to be a skilled warrior happens all the time in movies, all those training montages. Many DM's run D&D as a cinematic game, where training montages are more important than long tedious months or years spent in Wizarding school or Fighter Academy so you can multiclass when you level up.
The campaign world isn't built around game mechanics and people getting the fastest method of getting the minimal method of gaining a class. It is possible to learn a skill set quickly through on-the-job training, but it is often preferable (in ways not easily reflected in game mechanics) to learn it over more time in a more structured environment.
Or, to use another d20 game, think about Star Wars. Luke Skywalker multiclassed into Jedi Guardian not after being trained from infancy at the Jedi Temple, but after several hours of instruction from Obi-Wan on the flight between Tatooine and Alderaan. It was clear from what Yoda said later (and reiterated in the Expanded Universe) that Lukes training was incomplete and had serious gaps, but from a game-mechanic standpoint it was enough for him to multiclass to Jedi. In the later novels, Luke spends years filling in the gaps in his initial training, tracking down what he can to find the teachings that Obi-Wan and Yoda didn't have the time to pass on to him.
Gaining a Wizard or Monk level after a few days of practice is just like that, you are gaining the absolute rudiments of the class enough to be represented by that in rules purposes, but for roleplaying purposes you are seriously lacking in many elements of your training.