I do not like the idea of classes having limited spell list or having multiple full casters.
one fullcasting class is enough, with access to all spells.
but,
spells known are at the level of 2014 sorcerer. start at 2 spells and up to 15 spells at levels 17+
suclasses can give extra spells known as usual. depending on the subclass.
and you can exchange one spell known at level up.
this gives you freedom to make your mage anyway you like it with selection of spells, but the amount that you can bring to the fight is still limited.
example of subclass spell additions to spells known:
Healer;
cantrips; resistance, spare the dying
level 1; cure wounds, healing word
level 2; lesser restoration, healing spirit
level 3; mass healing word, revivify
level 4; Aura of light, death ward
level 5; mass cure wound, raise dead
Necromancer;
cantrips; chill touch, toll the dead
level 1; false life, inflict wounds
level 2; blindness/deafness, ray of enfeeblement
level 3; animate dead, summon undead
level 4; blight, shadow of moil
level 5; dance macabre, enervation
Kineticist;
cantrips; eldritch blast, mage hand
level 1; magic missile, shield
level 2; levitate, kinetic jaunt
level 3; fly, haste
level 4; Mordekainen's faithful hound, Otiluke's resilient sphere
level 5; animate objects, telekinesis
Traveler;
cantrips; guidance, mage hand
level 1; expeditious retreat, longstrider
level 2; misty step, vortex warp
level 3; thunder step, gaseous form
level 4; dimension door, find greater steed
level 5; passwall, teleportation circle
Greenseer;
cantrips; primal savagery, thorn whip
level 1; entangle, fog cloud
level 2; spike growth, pass without trace
level 3; plant growth, summon fey
level 4; guardian of nature, summon elemental
level 5; insect plague, wrath of nature
Pyromancer;
cantrips; firebolt, greenflame blade
level 1; burning hands, hellish rebuke
level 2; scorching ray, heat metal
level 3; fireball, ashardalon's stride
level 4; fireshield(fire), summon elemental(fire)
level 5; immolation, summon draconic spirit(fire)
Mindbender:
cantrips; friends, mind sliver
level 1; charm person, dissonant whispers
level 2; hold person, suggestion
level 3; enemies abound, fear
level 4; charm monster, phantasmal killer
level 5; dominate person, modify memory
Illusionist;
cantrips; message, minor image
level 1; disguise self, minor image
level 2; invisibility, mirror image
level 3; hypnotic patter, major image
level 4; greater invisibility, hallucinatory terrain
level 5; dream, seeming
Defender;
cantrips; bladeward, sword burst
level 1; absorb elements, armor of agathys
level 2; aid, blur
level 3; counterspell, dispel magic
level 4; banishment, freedom of movement
level 5; bigbies hand, dispel good and evil