In my opinion the quests are going to be BIG in this and coming metas, and that's probably not a big or risky prediction. Everyone knows how big of an impact the previous quests had and even though these new ones are lesser in power(by design) they still will have major implications in deck building(well most of them anyway). Here's what i think about them going forward.
Paladin: Starting with the one i think will have less and less impact as we go on(assuming they never print any more reborn cards after this expansion) in terms of deck design, there will be a "druid-ramp-like" package for the paladin quest of the best cars that have already been exstablished already and then you just fill in the gaps after that, the same way druid did in previous metas where they have the oaken summons, malfurion, spreading plague, ramp, packages and they just slotted in different win conditions like maly or hadronox. i think the paladin quest will be just like this in the future, and it will be locked to aggro generally.
Warrior: From now on in the list i'm going to work from left to right on the class menu but it's really coincidental that warrior is next after the special case that i brought up paladin first, since i think it is exactly the same as the paladin quest in some respects. But i think it is better because there is more likely to be more weapons in the future to change up the quest dynamic for deck building, but i still think in the end it will mostly revolve around people finding what works and setting up a "package" just like with paladin, maybe some decks will run said "package" with rush cards and make it more mid-range, or maybe it'll be the "package" and taunt cards, for a more value oriented way with the hero power, and play it off as jaraxxus but you summon 4/3s. I am interested although i don't have a lot of faith in this quest.
Shaman: This quest is getting the most hype and i think for a good reason, this quest has a lot of workaround for it since unlike weapons or reborn minions there is a lot of work-around that can be done, there are already decks for value and decks using the hero power for damage(but mostly you'll just see unimaginative lackey shit with it), and i think this will get more and more diverse as expansions go on, good design of a quest and i don't think the double battlecry reward is as busted as people might think at first glance.
Rogue: Okay this one i think deserves attention because i think in the coming days it will be VERY OPPRESSIVE. If you remember pirate warrior you'll know how dangerous a 2 mana fiery waraxe can be, and if you can get that guaranteed every turn, it's very dangerous indeed. Used to be with pirate warrior you'd just run them out of stuff and if you survived at that point they would just concede because they have no way of coming back. But with this new lackey shit, the deck has a way to make a full board out of nowhere and having guaranteed 3 damage to face every turn(this isn't a taunt heavy meta anymore remember that) it's pretty gross, and i foresee in the future it will be made for that oppressive nature, since the quest doesn't require deck building for the quest, since it's so easy to complete. Remember kingsbane before the leeching poison nerf? Kingsbane decks before the leeching poison nerf weren't aggro, they were focused on outlasting their opponents via the lifesteal from the weapon and trying to build the weapon as big as possible, but after the nerf to leaching poison that way of play was gone so the deck became pure aggro now relying on getting as much damage in as possible as fast as possible and i think that is the future that awaits the quest for rogue, and it's a pretty disgusting future indeed. Some people are trying to parade this is a "Control Rogue" enabler, but what can a 3/2 weapon do against lackey shit that fills a board every turn, and as a class that doesn't have boardclear it's just a joke, and it's even more of a joke against other late game oriented decks. If the quest didn't have such a low requirement then maybe it wouldn't turn to pure unadulterated aggro, but it doesn't.
Hunter: The hunter quest is very interesting to me, since token druid exists right now and has been in full swing since before this expansion i'm surprised no one has talked about it that much. I imagine soon there will be a "Token Hunter" if you will, and unlike druid they'll have 2 mana savage roar available at their finger tips, not as oppressive as the rogue quest in my opinion since the quest takes longer to complete and a deck has to be made for it to succeed, but i think it is a sleeper possibility in the future.
Druid: The druid quest is something a lot of people have been calling very versatile, since that is what the choose one cards are designed to be for after-all, and i think that's a good way of putting it. I've seen a couple decks here and there already since the expansion dropped and i have to say, bias out of the way this is the most interesting quest for me, i've seen heal druid decks with quest, and aggro type druids with druid of the scythe, i think the payout for the quest isn't too bad either unlike rogue, do nothing essentially for 4 turns and after that you can get more tempo on turns than normal seems simple enough. Although i do not enjoy the fact that it can be completed earlier if you have the coin, but i guess the innervate class doesn't really need that if they so desire.
Warlock: If you've seen me around you've probably seen me call plot twist warlock a meme once or twice, and although i use it as more of a joke most of the time it isn't inherently untrue. I think betrug is a joke and 8 healing MAYBE off brood mothers isn't reliable enough, if warlock had a way of going infinite with it sort of like dead man's hand warrior i think it would have been a more interesting deck as least for me, but that said i don't think the quest is good, people keep saying everywhere you go that once you get the quest completed that every card you draw is zero mana, but more likely than not, every card you topdeck is 2 mana is a better way of putting it. Since most plays you tap for if you tap you'll play that turn if only for the tempo of it, and i think combo decks with this would be too slow and unreliable so i don't know where this will go in the future.
Mage: I am designating this the "Fun Quest" of all of them, although i imagined it in a cyclone mage deck as an extra card generation engine i think it's more there for people that enjoy the randomness of hearthstone than the competitiveness of it, i've played against a couple quest mages with my singleton hunter and by the time they got the quest done they were usually dead, and my singleton hunter isn't even aggro. This is a hopeful yog's puzzlebox generator in most decks, and if you're not playing it for fun i don't know why you're playing it.
Priest: has a high chance of being oppressive in the future but no like lackey shit is, in more of an arena style of oppression where they just play big stuff every turn and hope to win eventually when their opponent has run out of things to play against them. As one of the few people that enjoyed dinomancy it's cool seeing it back in an actual playable form, and it's pretty cool that it's going to be meta for once, in the future i envision a lot of mid-range priest decks, that run a lot of healing and use the quest as a finisher rather than rushing the quest out like other classes, kind of like galvadon paladin of old.
All in all i like the new quests, although i think they are not all created equal in terms of strength like the older ones were. And although they are not as powerful as the older quests were i think that's a good thing, the meta needs to slow down a bit, the game is not just KFT and KnC it can't just be big fancy blowout decks every expansion. I think most of the quests are a great addition in terms of deckbuilding and although most people will just netdeck what is popular and don't try anything new i think the theorycrafters will enjoy messing around with them.
