Magical training is detailed in the APPRENTICES supplement.
The Gift can manifest itself at any time from Birth but the average is from about eight to about fourteen. Detecting if a child has the Gift is left a bit handwavy in the system (a spell that reliably worked would be a boon) but Ex Misc magi of the line of Pralix have a Supernatural Virtue (usually used to detect Hedge Magicians) that helps them in searching for apprentices too.
A Magus finds an apprentice (or if he’s a Bonisagus he can steal it from another magus!) and takes him in. This can sometimes be done with the approval of their families but sometimes it amounts to kidnapping. It can be terrifying to the child: remember they have no protection against the baleful effects of the Gift unless a Mage temporarily extends their Parma Magica over them. They have been condemned to spend years with a person their subconscious is telling them is an utter villlain and about to do terrible things to them. It’s a wonder magi turn out as well as they do. (This goes double and triple for Tytalus apprentices.)
The apprentice must acquire enough Latin to read the books he will be reading and will also be trained in the Artes Liberales and Philosophy. But the magical part of their training begins with Opening The Arts, which focuses their Gift on the path of the Hermetic Mage and gives them a rating in the Forms and Techniques. (The Gift can be opened to other disciplines but you can’t learn two different paths. Each path of magic has it’s own form of Warping BTW. Hermetics suffer from Twilight as the Magic Realm overwhelms them. The Pictish Gruagachan, for example, turn into trolls.)
The apprentice continues training for at least seven years. He has one season a year at least of formal instruction and also learns as he assists his parens in the laboratory. He learns Magic Theory and starts to pick up actual spells. He learns any extra skills that his House requires.
When his Master thinks he’s ready he is given one last test, the Gauntlet, which is of a form the master chooses. If he passes he swears the Oath is instructted in Parma Magica and becomes a mage. If he fails he gets to try again in a year. After three failures he has a right to ask for a different examiner.
Now, as for becoming a Quaesitor when you’re not a Guernicus this is possible but it would require you to make your character several years older than a just Gauntleted novice magus. They would have to impress the Quaesitores with their suitability to be an investigator and then undergo training, possibly as long as another apprenticeship to gain the needed skills. Starting as an older mage is possible… But what do you see the advantage of not being a bog-standard Quaesitor of Guernicus to be?