There are few colleges of magic and no standardised system of teaching magic, so each master is a little different. The description below is what a typical apprentice might face.
Magic is not an easy craft. It requires incredible mental fortitude and training to comprehend, and a large breadth of background knowledge. A new apprentice will typically first be taught to read and write Common if they are not already able to. Then they will spend some time doing simple studies of history and natural philosophy, imparting them with broad knowledge of many intellectual matters that are necessary to be an effective mage. These studies will continue throughout most of their apprenticeship, but after up to 6 months of dedicating themselves wholly to these studies they will begin their training proper.
After this introductory period, a new student's first real tasks involve mental exercises; often going on for at least 3 months before any further studies are added to their workload. They are trained to conjure up complex patterns and solve logical puzzles within their mind, holding them in place; the goal of this is to strengthen their mind and improve its capability to think in ways to which it is not accustomed. As their training continues, these will be supplanted by more advanced trials, requiring the aspirants hands, mouth, and mind to work seperately and maintain their concentration even in the face of distraction and pain.
As time goes on, new studies are added to their current exercises, which continue in order to strengthen their minds. Usually a year or so after the commencement of their apprenticeship, the aspiring mage will begin to learn the letters, pronunciations and grammar of the arcane tongue and the nuances of its alphabet. They will be drilled relentlessly, for even the slightest mistake in pronunciation can ruin a spell or have unintended effects. After a while, they will also begin studies in drawing - not art so much as perfect reproduction - until they can perfectly replicate magical symbols and sigils without the slightest error. All this continues alongside their other studies, with mystical treatises and works on magical theory largely supplanting the study of history and philosophy.
Eventually, usually after about 2 years of study (but possibly after as long as 4 years), the apprentice is ready in skill and prepared in their mind for what is to come. At this time, their master will usually allow them to begin studying of the simplest spells, known as cantrips. These spells are so trivial that an experienced magus may cast them without even needing to prepare them, but for the aspiring mage the mental calisthenics required surpass anything they have wrestled with in the past. These cantrips have the same scope and limitations as those detailed in the PHB under the first level spell, Cantrip. Once they have mastered these simple workings and shown their ability to manipulate the forces of magic - however minute - their arcane training can truly begin.
Their study of cantrips continues, and once they have mastered them, their master will begin to introduce them to true spells of the first level. All of the mental exercises and training have led up to this; besides the complex symbols and sigils that must be fixed in mind, magic words that must be perfectly pronounced and gestures that must be made without error, a spell has a tangible presence, almost a sentience. Committing even a weak spell to memory is a mental battle of wills, struggling to bend the power of your mind and pin down the words and symbols of power, fixing them within your mind to be released at a later date. Depending on the intelligence and dedication of the student, the memorisation of their first true spell can take anywhere from a single week to months and months. From the first time an apprentice manages to prepare and cast their first spell, they are considered to be a 1st-level wizard, starting at 0 experience and being able to prepare and cast a single first level spell per day.
From this point on, they are effectively a first level wizard, though if they leave without further instruction after their first successful casting they will gain a 50% penalty to their first 1000 experience points and their spells will take twice as long to prepare until they have gained 500 experience. At this point, they usually become the master's personal assistant for at least 6 months and up to several years - or possibly even for the forseeable future. Their formal studies in symbol-drawing and their mental exercise sessions are dropped, although they are expected to keep up practise themselves in order to maintain their edge. They spend their own time to honing their newly-found art and practising the magic they have been taught until the master feels they are ready for another spell. They also tend to act as the master's assistants, teaching younger apprentices and doing tasks for their master. The main advantage to staying on with the master after gaining magic is the promise of additional spells, as their repertoire will be very weak indeed if they leave too soon. They also get additional tutelage in how to use and control their power and thus avoid the 50% experience penalty if they try to "figure it out on their own".
It usually takes between 3 and 5 years for an apprentice to go from complete novice to becoming a 1st level mage, and most apprentices will stay in their master's service for at least another year besides. In addition to gaining the ability to cast cantrips, level 1 wizards begin with a spellbook that contains Read Magic and Detect Magic as well as 2d4 other first level spells. Usually upon setting out, level 1 adventurer wizards will start with a small stock of the spell components necessary for their spells; up to 10 castings of components that are common and worth 1 gp or less, or 1 casting of components worth up to 100 gp.