Mulligans
General Mulligans
You always want the quest in your starting hand. Besides that, Thalnos is always a keep for carddraw or for spell damage with Spirit Lash and Shadow Visions is a good keep to get specific spells for specific matchups.
For agressive opponents you need health gain and removal, for others you mostly need card draw.
Priest
I got an idea after seeing the new Reckless Experimenter card reveal.
Reckless ExperimenterYou can combo the card with Gilded Gargoyle and Cube to gain a lot of coins without spending a lot of mana. These coins can be used in with velen and mindblasts or with shadowreaper anduin to deal lots of facedamage in one or a few turns.
Dead RingerThe newly revealed Dead Ringer card can contribute to more consistent combos due to the amount of draw you have available in priest now.
Expendable cards are Umbra (I thought it might be handy in a combo) and Archbischop Benedictus (I just love the card, but it might also be good if your opponent has to much health and you need to win the fatigue game), maybey add Alexstrasza or Chameleos instead. You might also want to switch Rotten Applebaum for something like Voodoo Doll or Power World Shield.