It is a bit tricky with warhammer fantasy, as most of the factions qualify as core factions.
However, the big 4 to me are;
1. Empire. They are the most significant and central faction in the warhammer world. They serve as your everyman human faction, and are the "good guys/main faction". They have a strong focus on combined arms tactics, using disciplined infantry formations backed up by massed gunpowder artillery, supported by elite cavalry.
2. Warriors of Chaos. They are the main chaos faction, and one of the main villains of the setting. They contrast well with the empire, as they have strong, heavily armoured troops, unwashed barbarian hordes, big monsters and a bigger focus on melee, while the Empire uses lightly armoured, disciplined troops with a focus on defense and shooting.
3. Orcs & Goblins. They are the most prominent of the "neutral" factions- They are not aligned with Chaos or the Undead, but still evil. They are everywhere in the warhammer world and they fight everyone. They are crude, rowdy anarchists with lots of crazy (but disfunctional) weapons and monsters.
4. High Elves. They are the most prominent/popular of the good factions besides the Empire. They have more fragile, but faster/more skilled infantry than the other factions, with a big focus on archers, cavalry and magic. They contrast well with the Orcs & Goblins.
Honourable mentions go to;
The Skaven. Like the Orcs, they are pretty much everywhere and fight everyone. They have the weakest, most cowardly troops out there, with a big focus on being sneaky, underhanded and backstabbing. They have the most advanced technology, using a mixture of magic and warpstone fueled steampunk to create bizarre weapons. They focus on using vast hordes of slave infantry backed up by weird weapons and monsters.
The Dwarves. They are the 3rd of the big 3 good factions. They are one of the slowest moving armies, with very durable, stubborn troops that are very hard to kill or break. They don't use magic at all, so contrast well with other armies in that regard, and are very good at countering enemy magic. They contrast very well with the skaven, as the skaven and dwarves are mortal enemies.
Those 6 factions are the ones I would see as most likely to be included in any warhammer game.
For expansion packs, I would go with the Vampire Counts & Bretonia for one expansion pack, and Dark Elves & Lizardmen for another. After that all you have to fill in is Wood Elves, Beastmen, Daemons and Ogres, who are probably the 4 least prominent factions in my view.