Yay! A fellow (potential future) Cthulhu enthusiast!
Keep in mind that the 1-ofness is the reason why you don't get to see 7 factions in a single core set (the only reason why you get to see the 8th is that it was added later, in a deluxe). It sorta blows that you need 3 cores to have a full playset of stuff, but then again, the same is true for all LCGs save SW.
Pretty much, due to the combination of direct removal (Short Fuse, Shotgun, Shotgun Blast) and small/cheap characters with Investigation. There's even some Willpower to be found on reasonably-costed dudes. Another solid core-set faction is Syndicate - a lot of their main tricks are featured in the core, there's decent characters and a crap-ton of readying/exhausting tech. Where Agency aims to push stories through bullying/damage, Syndicate goes for exhausting would-be defenders.
Out of the monster factions, Shub-Niggurath is probably the most solid, not least due to having Y'Golonac - that thing is extremely hard to handle in a core-only environment (it's pretty much Deep One Assault or you're done ).
If the core set piqued your attention, I'd say the next purchase should be Secrets of Arkham - it's sort of like a mini core-set, in that it has an even number of cards for each faction. It also has a new set of stories you can play with when the core set ones get stale for you.
Watch out though, the old editions of the box only had 2 of each card, rather than 3, so if you're getting it, check that it has 160 cards (3 copies of 50 new cards, plus a new 10-card story deck). The old ones probably shouldn't even be in circulation anymore, just mentioning it to be on the safe side.
The box as such strengthens the core set themes of each faction, so it's a natural first buy. After that, I'd say a second core is best (as there's a lot of very good cards in core, similar to Netrunner). Then, you probably just want to grab the deluxe of the faction that most interests you. Some random notes on the boxes:
- Each of the faction boxes also includes 2 cards for each other faction, so you're never just getting one color
- You probably want to stay away from Order of the Silver Twilight initially - it's a box that adds an eight faction, which is actually very tricky to build and play properly (and probably needs some of the newer cycles to be even functional). You'll still get a handful of Silver Twilight cards in the newer boxes, some eminently splashable (and you only really need like 9 cards from a faction to be able to splash it, really )
- Also leave Terror in Venice for later - it's mostly focused around Day/Night mechanics and Conspiracies, which are two levels of complexity you don't need right now. There're some interesting cards there, but overall I'd say it can safely be one of the last purchases.
- All the faction boxes are rather self-contained, you can always build a decent deck out of one (or two) core(s) and a faction box.
- The factions most helped by their deluxes are probably Miskatonic University (Seekers of Knowledge) and Cthulhu (The Sleeper Below), Syndicate (Denizens of the Underworld) also has a powerful deluxe.
Seekers of Knowledge is a good buy if you're interested in seeing some more of what the game has to offer mechanically - namely, there are Prophecies which are events you play on top of your deck, and if a particular trigger happens before you draw that card, you get a strong benefit. Each faction has a prophecy tying into its strengths.
- Agency's deluxe (For the Greater Good) is also very strong, adding some needed trickery to the faction while managing to stay true to the original feel from core. If you like Agency, it's a no-brainer.
- Yog has a weird deluxe - it's one completely self-contained theme (Yithians) that ended up so strong that it's broken up by the restricted list, plus a handful of individual interesting tricks that open completely new avenues (check out this spell, for instance - you can pull off some positively insane stuff with it). They also tried to open a new theme for Yog (namely Sorcerers), but the support wasn't quite there. There's some more combo pieces in later deluxes that help with it though.
- Shub's deluxe just came out and mine hasn't arrived yet, so I didn't fully digest it (but from the spoilers it seems that it has really interesting cards, both for Shub and for the others). The new cards there will particularly help Hastur, actually.
- Hastur's deluxe is the next scheduled release, coming this fall or so.
That's enough for now I think, if you want to know anything specific just ask!