D&D 5th Edition’s (#dndnext) classes seem to be varying between 4 and 50. I think there only need to be 3, just like in Original D&D, or Swords & Wizardry White Box.
Fighting Men: The Fighter fills every non-magical combatant role from heavy knight, to barbarian, to hunter, to thief. Which weapons and armor the Fighter uses determine what else they can do. A knight in full plate and a half-dozen heavy weapons needs a heavy mount to get around, and the expense of this implies either a feudal system with dozens or hundreds of peasants supporting the knight, or massive treasure hoards protected by dragons, that only a knight can extract. A barbarian’s few heavy weapons and medium armor make movement important, and probably sailing or riding to strike with surprise at undefended enemies. A thief uses light weapons and light or no armor to get maximum speed and stealth.
Magic-User: The essential traits of a wizard, from Merlin to Thoth Amon to Elric to Gandalf, are direct attack spells, and information gathering: Detects, identify items, ESP, clairvoyance, and languages and lore skills. I’m no fan of Vancian magic, but for D&D it’s traditional and mandatory. To make up for the very limited low-level spells per day, I use the cheap scroll-making system from original D&D (100 GP/level, 1 week/level), and gift starting Magic-Users with a few 1st level and maybe one 2nd level scrolls. An alternative solution is to have cantrips which can be performed for free; this makes Magic-Users more true to genre, and a weak attack cantrip (1d4 damage at most) or Harry Potter-ish Stupefy and Disarm charms keep them in the game. Magic-Users in myth and literature wear no armor, but often have weapons, especially swords.
Cleric: Clerics cover the range from pure spell-using priests to combatant paladins, Brother Cadfael to Roland. Holy men healing people is a common fantasy trope, all the way back to that weird “Bible” anthology some people take so seriously. Turning the undead makes Clerics useful even when their spells run out. Scrolls can fill out low-level healing abilities, as with Magic-Users. Clerics wearing heavy armor is unusual, but some paladins like Roland and Turpin did; the old “blunt weapons” restriction only makes sense for one segment of priests of one religion at one point in history, and should be discarded.
Not Included: Thieves are just very weak Fighting Men, they do nothing special, if you have even the simplest skill system or stat checks. Rangers are just Fighting Men with bows. Paladins are just Clerics who fight more than cast. Cavaliers are Fighting Men with horses. Assassins are just Fighting Men; killing people is what Fighting Men do. Druids are just neutral Clerics.
My thoughts on races may be seen even more radical, but that’s because they are based on history and mythology, not the incestuous Tolkien-D&D-novelization-pseudofantasy cycle that ends in Salvatore’s “Good Drow” bullshit books.
Dwarfs: In Norse mythology, Dwarfs were maggots in the body of Yggdrasil, and only mentioned being powerful Fighting Men or having dark magical power. Wagner’s Ring Cycle reinforces these roles. Tolkien’s adaptation is peculiar for having no practical magic, but their role as heavy fighters was set. I’d prefer if they could only be Fighting Men, but wizened Dwarf Magic-Users like Alberich are appropriate. Is it too late to dump the anti-magic nonsense?
Elves: In Norse mythology, the Alfar are either of light or darkness, and do healing or cursing magic depending. In Celtic mythology, the Elves do mostly illusions and sometimes healing magic. In Tolkien, Elves do healing magic, there are no Elven battle mages, that’s what they use bows for. So I’d prefer if they could be Fighting Men or Clerics, and not Magic-Users. Certainly I see no precedent for the multi-classing element.
Other Non-Humans: Genocide. Hobbits, er, “Halflings”, are vermin, an abomination to be wiped from gaming. Gnomes are annoying, best used as nuisances by the DM. Half-breeds aren’t as good or interesting as their full-blood parents. Orcs, Ogres, Goblins, Lizard-Men, etc. don’t fit well in a Human-centric campaign, which most fantasy literature reflects. These could well be statted up in the Monster Manual so the DM can allow them in unusual campaigns, but in the main book, only Humans, Dwarfs, and Elves have any legitimacy.