Mulligans
General Mulligans
These cards are pretty much just some relatively powerful cards. Babbling Book offers some early game board presence and a random spell, that over 5 mana adds to Alanna, and under, sometimes offers some early game tempo. Arcanologist offers a decent early game stated body, plus a way to draw Ice Block, which let's admit, is a part of every Mage deck. Thalnos is Spell Damage for AoE, and card draw so what's not to like? Frostbolt and Primoridal Glyph may screw over the Dragon's Fury a bit, but they are just so good early game spells and flexible late game that it's hard to keep them off. Intellect will help draw cards when you're down on some, and Fireball is just a good spell for removal and burst.
Aggro Mulligans
Not so much aggro, as I prefer to use these as removal, but Frostbolt and Fireball do offer some reach if you need to get lethal.
Midrange Mulligans
Shimmering Elemental is a decent draw to contest the board, plus it's an elemental and a one health minion for Jaina to ping off. Courier adds some value later on, but early on can offer some board presence. Tar Creeper and Water Elemental are a little better at that but it's still decent. Kazakus just sort of fits in a Kazakus deck, and he does provide a decent way to either clear the board or get more value. Harrison to get rid of Kingsbanes and Skull of Manaris if your having trouble with them, while also providing draw. Same notes on Volcanic Potion and Fireball.
Combo Mulligans
Same notes on Artificer, Arcanologist, Kazakus, Harrison and Jaina apply here. Medivh offers some value on the higher end spells, expecially the Kazakus 10 mana potion. Sindragosa offers two Water Elementals essentially, and some bonus value legendaries which range from whatever, to decently good. Alanna is Nether bait or a finisher. It depends on what the board state is like and actions leading up to her play.
Control Mulligans
As this is a control deck, sustainability is vital. That's why Artificer and Jaina are in the deck. Jaina gives the elementals like Water Elemental and Tar Creeper some lifesteal shenanigans, and Artificer makes it so you can heal 7 for casting Flamestrike. Doomsayer is good in early game, but not very consistent later on. Try to mulligan for it in an aggro match up. Tar Creeper is much of the same, but with Taunt it gives you some nice life. Volcanic Potion clears some of the really small Silver Hand Recruits and other early game minions. Polymorph for other control decks like Big Priest and Control Lock. Dragon's Fury seemed pretty good for a big board clear, and it mostly works if you mulligan for the early game spells. Flamestrike is a lot more consistent and won't kill your Artificer after being casted.
A fun little Control-Kazakus Mage I made with what I had. It’s a whatever deck, no real competitive viability past Rank 15, but it’s fun enough.