I see the three core class as fighter, mage, thief. With a slight tweak to the magic system, a cleric is just a mage.
But really the three core class are defined by the different pillars: combat, social, exploration. So fighter for combat, skill guy for exploration, face for social. However, in D&D the face is just a Skill Guy with a Social focus. The mage is just a class that on the fly be useful in all three pillars.
Then you mix and match to get your classes, which basically become.
Core Class + Archetype + Background
Core Class A + Core Class B + Archetype + Background
Where Archetype is how you do your class and background is how you became your class (as it were).