Mulligans
General Mulligans
Mulligan for at least a Flamecannon, Forgotten Torch or Frostbolt if possible. Against any class with weapons, pick Snowchugger. Flamewaker for Zoos/Aggro. Important: it is crucial that you ALWAYS protect your vital momentum-stopping minions (i.e. Snowchugger/Flamewaker) with a Counterspell or hide it behind a Duplicate (make the target obvious).
Combo Mulligans
Sorcerer's Apprentice will make A LOT of your spells cheaper, and give you opponents a cheap, non-essential target to burn, in turn gaining card advantage as well.
Echo of Medivh is to be used when you have 2-3 important minions you want to copy. Do not waste it otherwise!
Patience is the key and thus, do not hesitate not to finish your mana (rather than to use a card non-optimally).
Grow your Ethereal Arcanist(s) with secrets and you'll finish your opponents in no time.
So far the only problematic opponent(s) I'm getting are fellow mages. Be patient with these guys and always protect (or even better, double protect) your crucial minions.
Break your opponent’s early dominance and board control/synergy with Flamecannon, Frostbolt or simply using your trusty hero power! Use your cheap Snowchugger to freeze the opponent’s face or Flamewaker plus more cheap spells to clear board. Attack face with Animated Armor, Ethereal Arcanist or even Emperor Thaurissan if it doesn’t get killed too quickly. Opponents will usually run out of spells to drop all your minions since each of them are so useful. Patience if your best friend.