This is what let me go from diamond 5 to legend very fast. Its burst potential is really high and though lack of Elemental Evocation hurts, it can still go off quite fast.
Elemental Allies - best early draw for mage which nobody plays, because nobody plays enough of elementals to justify including this card. We play 7 elementals and it's enough most of the times.
Learn Draconic - early game 6/6 is great, try to finish quest with Khadgar on board or mana open for Conjurer's Calling - two 6 mana creatures are much harder to deal with than one. Watch out trying to finishing quest with Sorcerer's Apprentice out - a bit of anti-synergy here.
Ancient Mysteries - secret package gives us early game survivability, healing for 8 and making these 3 mana cards cost 0 helps us using them to go off. Thining the deck is also important.
Astromancer Solarian - does not directly contribute to any of this deck's themes, but it's just good to have something to play early and value from prime version is very often just incredible.
Sorcerer's Apprentice - glue that holds this deck together. Mana cheating is absolutely essential with this deck, you need her to go off with Chenvaala or Mana Cyclone, also getting Mana Giant to 0 cost as soon as possible. I had a game with turn 2 unanswered Sorcerer's Apprentice, which turned to 3 5/5's next turn.
Arcane Intellect - early draw is very important for this deck, you need to gather cards for going off
Chenvaala - great combo with Sorcerer's Apprentice and cheap spells making other cheap spells. Also nice synergy with Elemental Allies - you want to play Chenvaala as second elemental to get more spells -> more elementals.
Conjurer's Calling - one reason for why Tempo Mage is underrated is that it rarely plays this card. Game winning with Mana Giant, Khadgar, you can also use it early with Learn Draconic or Chenvaala elementals.
This is a deck of few different combos and really high burst potential - sometimes it looks like you get massive board out of nowhere - timing Learn Draconic with Khadgar and Conjurer's Calling for example. There is quite high RNG factor, though not as much as in Casino Mage, so you need to adapt to different spells you get from Mana Cyclone, Evocation, Solarian Prime or creatures from Conjurer's Calling.
For starting hand keep always keep draw and Sorcerer's Apprentice. Sometimes you can keep Chenvaala (with Sorcerer's Apprentice and cheap spell(s)), cheap elemental (with Elemental Allies), but only draw and Sorcerer's Apprentice are 100% keep.
This deck is fast enough to compete with Demon Hunter, Hunter and Warrior, has high enough burst potential that Rogue cannot clear board after we go off. Winning with Priest requires resources management and getting as much value as you can - but sometimes even Priest cannot find right answer for early burst. Leveling from diamond 5 to legend I met no other classess so cannot say how this deck performs against them.