I was going to mention earlier that some of the earlier Fulci films - like Lizard in Woman's Skin, Don't Torture a Duckling and One on Top of the Other - are very good examples of the kind of slightly surreal, mood-oriented films that fits in with the type of thing Franco was doing. The later zombie films are also, um, "narcotic", for lack of a better term for it, in their atmosphere, and definitely don't adhere to the normal plot conventions, but they're definitely worth checking out. And of the later ones, I really like The Black Cat as well, with Patrick Magee ("Victim of the modern age!!!")
Of Fulci, I’ve seen City of the Living Dead, and Zombi, which I wasn’t much of a fan of, but I read his Giallo are better. Also plan on watching his films this October on Shudder.
Fulci is interesting in that he has a career spanning several decades but only has a decade or so of his work discussed all that frequently. I confess I share those blind spots, but outside of his horror films I quite enjoyed Four of the Apocalypse, which has some of the same hallucinatory quality and violence as his best known films, but with a disarmingly melancholy tone.
If you've seen City of the Living Dead, you'll know roughly what to expect with The Beyond (which is probably his best movie), but I think it's a nicer looking movie (largely thanks to the aspect ratio) and a less sparse one as well, narratively. I do think comparing these two to his giallos makes a clear case for the loosening narrative structures as a deliberate artistic choice, necessary for the nightmare logic both films pursue.
Zombie I didn't gel to at all when I first watched it, but once I got used to how his movies...moved, I found it played a lot better. Some very nice jungle atmosphere, which I admittedly have a weak spot for.
JJ made a good case for his early giallo, but I think New York Ripper from his later period is worth your time as well. About as mean-spirited as I've seen a movie be, but undeniably stylish, Donald Duck voice be damned.
As for other essentials...if you have the room, I'd make time for Cat in the Brain, which has the man himself taking a starring role. It's not quite as pointed a comment on his work as Argento's Tenebre, but plenty of fun for fans. Might make a good double feature with the doc Fulci for Fake, which provides some interesting personal context for this and New York Ripper especially.