Magical artists are seemingly able to create magical paintings with the personality of a person, suggesting a great deal of magic ability and skill.
Where exactly is this skill taught? Are there magical apprenticeships or a college perhaps?
The Pottermore page for Hogwarts portraits gives a certain amount of insight into the process behind creating a magical painting.
Hogwarts portraits are able to talk and move around from picture to picture. They behave like their subjects. However, the degree to which they can interact with the people looking at them depends not on the skill of the painter, but on the power of the witch or wizard painted.
When a magical portrait is taken, the witch or wizard artist will naturally use enchantments to ensure that the painting will be able to move in the usual way. The portrait will be able to use some of the subject’s favourite phrases and imitate their general demeanour.
("Hogwarts Portraits", Pottermore).
This tells us a number of things. Firstly, there is such a thing as a "wizard or witch artist". Whether this is a distinct, formal profession or not isn't clear. It's possible to be an artist without being a professional artist. Nevertheless, the terminology exists. Secondly, enchantments are involved in allowing the portrait to move and talk. It would be reasonable to conclude that more proficient artists would be able to refine this process to create a more sophisticated final product. There is an element of artisan craftsmanship involved. Thirdly, creating a portrait has as much to do with the subject as it does the artist. A portrait of a goat won't be the same as a portrait of a famous witch, no matter who's painting it.
When an artist 'signs off' the portrait by giving it to the subject then it is not necessarily complete. The portrait can still be trained by the subject to make its character and personality more refined.
Traditionally, a headmaster or headmistress is painted before their death. Once the portrait is completed, the headmaster or headmistress in question keeps it under lock and key, regularly visiting it in its cupboard (if so desired) to teach it to act and behave exactly like themselves, and imparting all kinds of useful memories and pieces of knowledge that may then be shared through the centuries with their successors in office.
("Hogwarts Portraits", Pottermore).
The creation of a portrait can therefore be seen as a joint contribution between the artist and the subject. This dilutes somewhat the concept of a standalone professional artist.
All in all, it isn't clear how magical artists develop their craft. They clearly have a role in painting the portrait in the first place and in casting enchantments on the portrait to make it come alive. However, whether it was a professional vocation or simply a hobby taken up by enthusiasts isn't apparent. We never see anyone painting a portrait in the books. Similarly, when careers are being discussed in book five magical art school never comes up. And whilst we see what appears to be a professional photographer in canon we never witness anyone who could be described as a professional artist.
A paunchy man, holding a large black camera that was smoking slightly, was watching Fleur out of the corner of his eye.
(Goblet of Fire, Chapter 18, The Weighing of the Wands).
My best guess is that it's a skill which can be learned from a book. In the huge archives of the Hogwarts Library or of Flourish and Blotts you'd expect to find a book that can teach you the requisite knowledge. If you can learn how to care for a dragon or degnome a garden in a book then no doubt it'd be possible to get a book on art theory. An artist's craft and abilities would then be perfected and honed by years of practice. In other words, I imagine that people self-teach without formal instruction or tuition.
I don't think there is any reason to believe that magical, animated paintings are in any way fundamentally different from the similarly animated photographs that are commonplace in the wizarding world. In Harry Potter and the Chamber of Secrets, Colin Creevy—a muggle-born, first-year student—is excited that he has learned how to treat his photographs to make them animated. Rowling later indicated that, as she envisioned it, the use of a magical developer solution was all that was necessary.
I do not think there is any reason to expect that paintings are fundamentally different in this regard. So there would be no special skills or training needed to animate a portrait. If you use the right magical ingredients (mixed into your paints, or whatever the proper recipe is), you will get a portrait that can talk and move.