Playing this deck against an aggro deck is really easy, and you'll generally have a clear advantage. Just play out your curve and eat their creatures with your high toughness creatures. They should run out of steam, and because you're Priest and playing a lot of taunts you are a lot stronger against reach than most decks. You should crush most Hunter decks, most Paladin decks, and aggro Druid. Against Tempo mage it's all about getting board control first, never let their creatures survive if you can help it, their spells are not scary at all if they have no creatures on board.
Against combo decks like Patron you will want to go for the kill as fast as possible. By doing this you will force them into a position where they will have to use their combo pieces to deal with your creatures, then you can either outvalue them or go for the kill from there depending on your position. You need to be very aware of what cards your opponent could have and play accordingly. Against Warrior you need to watch out for Whirlwind effects and Execute. You have a lot of options between playing more creatures, healing creatures, buffing creatures, and drawing cards.
Against control you'll want to establish board control to force something. The control matchup is a very skillful one because it is going to be difficult to determine whether you should try to go face or outvalue the control player. It might often be best to go for outdrawing the control deck while maintaining a reasonable board position, rather than dump your hand where you could run into a board wipe. I've won many games where I've chosen to kill Sylvanas over going face, the important thing is to make sure you have board control at the end of every turn so that you have initiative, Ysera and Nafarian do nothing for your opponent if there is enough pressure (especially possible lethal) on the board.
Against handlock you want to stall them at 16-20 life depending on what your goals are for the next couple turns. Either construct a board that can kill them in one turn or go for outdrawing them while they sit around using either Clerics or Chromaggus.
The control dragon Priest matchup is by far the hardest for this deck, since they will only be slightly slower and your creatures won't be able to get through their's (low power, high toughness vs low power, high toughness). The match is almost always going to be won by either having an aggressive opening that includes Velen's Choosen or by either keeping a Cleric or Chromaggus alive (bait Shadow Word: Death with Crusher).
Mulliganing
Power Word Shield: Usually you don't want this in your opening hand, because it is basically a good card to draw at any time and you need a good curve. However PW:S can sub with a two drop or a hero power for a three drop, keep it if you already have a strong opener and are playing against a tempo or burn deck.
Twilight Whelp: Almost always keep if you have another dragon.
Faerie Dragon: Questionable against decks that can easily deal with it like Paladin (Minibot) and Warrior (Axe), but otherwise solid, especially with Shrinkmeister.
Wyrmrest Agent: Generally the two drop you want, almost always good to keep one.
Shadow Madness: Good against face and tempo decks.
Twilight Guardian: A decent activator for your earlier dragons and a very good turn four play. Keep it if you aren't too concerned with your early game.
Misc. Dragons: If all the other cards in your hand depend on having a dragon it might be worth keeping an activator. It's better to have a dead card that allows you to make great tempo plays with your other cards then a handful of mediocre plays.
Substitutions
Lightbane and Darkbane can be replaced with Blackwing Technician (I don't actually have a Darkbane yet, so I'm using a Techy). Chromaggus can be replaced with a second Azure Drake or possibly Nefarian or Ysera (though I have not tested them). Dr. Boom can be replaced with an additional tech card (like Shadow Word: Death), a Vol'jin, a Cabal Shadow Priest, or a Sylvanas Windrunner.
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