In response to a request from an online acquaintance, I made up a list of recommended reading. These are my own preferences! Also, Lovecraft is problematic in case you didn't know. The primary issue with his writing is both overt and subtly embedded racism. You should know that going in.
It seems to me that Lovecraft's works break down into three broad reading categories.
The Mythos – the core works that feature entities from Lovecraft's pantheon of horrors, typically an academic hero (archaeologist, psychologist, etc.) who follows a thread of investigation into revelation and madness. The Call of Cthulhu or Shadow over Innsmouth are solid examples.
The Dream Cycle – fantasies in the style of Lord Dunsany that are more focused on wonder than horror, often exhibiting a quest motif and, as the name indicates, occurring in a dream world. Celephais or The Dream Quest of Unknown Kadath are great examples.
Odds and Ends - including pulpy stories like Herbert West - Reanimator, 'science fiction' stories like Cool Air or From Beyond, scraps of stories, and stuff that was co-written with other authors.
Here are my favorite stories in publication order. I put an "+" next to ones I think are "must reads."
+ Dagon (mythos)
+ Beyond the Wall of Sleep (mythos, odds & ends, dream cycle?)
– The Doom that Came to Sarnath (dream cycle)
– The Statement of Randolph Carter (dream cycle)
Randolph Carter is a recurring character and loosely represents Lovecraft himself. The supposed chronological order of the Carter stories are: The Dream-Quest of Unknown Kadath, The Statement of Randolph Carter, The Unnamable, The Silver Key, Through the Gates of the Silver Key.– The Temple (mythos)
– Facts Concerning the Late Arthur Jermyn... (odds & ends?)
– Celephais (dream cycle)
+ From Beyond (odds & ends)
– The Nameless City (mythos)
+ The Music of Eric Zann (mythos)
– Hypnos (dream cycle, odds & ends)
+ The Hound (mythos, odds & ends)
Hypnos is good, but The Hound is similar and better. I included them both because they speak to the possibility that Lovecraft was homosexual. He was married to a woman at one point, but his marriage was an abysmal failure from the start. His life was given over to study, writing, and correspondence, not exclusively with men, but mostly with men. I personally think he was more "asexual" but if he were a sexual creature he probably would have been gay - certainly in our more liberal age he might have been.+ The Rats in the Walls (odds & ends, mythos)
Rats in the Walls and Facts Concerning... share the theme of a hidden horror in one's ancestry, another common theme for Lovecraft.– The Shunned House (odds & ends)
This one is notable because it has some of that pulp vibe that the Call of Cthulhu role-playing game has - fighting elder horrors with flamethrowers and such.– Cool Air (odds & ends)
+ The Call of Cthulhu (mythos)
+ Pickman's Model (mythos, dream cycle)
+ The Dream-Quest of Unknown Kadath (dream cycle)
Dream-Quest is ridiculous. :) I think I mean that in a good way. It's very "high style" stuff.– The Case of Charles Dexter Ward (mythos)
+ The Colour Out of Space (odds & ends, mythos)
+ The Dunwich Horror (mythos)
+ The Whisperer in Darkness (mythos, odds & ends)
+ At the Mountains of Madness (mythos)
+ The Shadow Over Innsmouth (mythos)
Shadow Over Innsmouth is the quintessential story about the "deep ones" and is genuinely scary to read.– The Horror in the Museum (mythos)
+ The Thing on the Doorstep (mythos)