So I got a Core set on a whim and my wife and I played the first scenario. We got instantly hooked and we've been buying up cycles of content (so far we've got all of Dunwich and Carcosa, as well as a second Core). Now we haven't dug into the cycles yet and my question is this:

Should I pop the packs open and take the player cards out in advance, or just play with what we have and open the packs and get the player cards as we go?

Buy and open everything

As the saying goes:

“Once you open a pack you can’t put it back.”

So do you want one big rush of maximum new player cards or do you want lots of mini rushes of “oooh, new player cards!”

It's sort of as your preference dictates. I'd open some packs and look at some cards together. You can discuss thier potential, or if they suck or whatever. When that stops being fun, or you start to get overwhelmed by options, then make some decks and actually play the game. Save the rest for later, when you are used to what you've got already. If you're opening packs like a mad fiend and the point of glossing over cards never comes and everything is still fresh and exciting from start to finish, then hey, bonus.

For anyone viewing this in future, since I imagine the original poster has made up their mind by now, I think opening packs by cycle is good mix of variety and not getting overwhelmed by the card pool.

So start with the Core Set, then when you are ready to start The Dunwich Legacy open the deluxe and all the Dunwich Legacy mythos packs but leave The Path to Carcosa sealed.

I'm about to finish Carcosa cycle, I've completed Dunwich.. And I'm opening them as I go. There's no real need to do that.. But I'm digesting the card pool nicely at this pace imo.. And there's still lots I haven't gotten to.

Ill start Forgotten Age with 2 new classes and build all over again... There's no wrong answer imo

We did the same thing. We bought all the way through TFA and opened the packs as we went. The good is that it’s like you starting at ground zero and get a feel for how different cards and game mechanics work. Plus every time you finish a scenario, it feels like Christmas! New cards. Yay!!
The bad is that you think you have storage figured out and you don’t!

With Asmodee no longer providing product support directly to consumers (you must return entire games to the retailer if cards/pieces are missing rather than requesting replacements from Asmodee), you should now always open and inventory everything immediately as you're out of luck if you open them later to find missing components.