Start with Call of Cthulhu if you really want to, it's an OK story but nowhere near his best and Cthulhu himself is overhyped.
My personal favorite is The Shadow over Innsmouth, it's written in a slightly more modern 'pulp' style (similar to Dashiell Hammett) and can elicit some real shudders. It also has a Twilight Zone/Shyamalan-on-a-good-day twist ending, so be aware of that.
Most of his stories are either first-person narration, either from the character experiencing the horrors or a secondary 'Watson' type figure. It's a good style, but he can overdose on the adjectives sometimes - take a drink every time he describes something as 'eldritch' or 'nameless', you'll be hammered by page 20.
Herbert West: Reanimator is also a good entry-level one, it's basically a zombie story but goes down some weirder paths than the stuff we see nowadays (god zombies are getting dull).
If you're interested in his trippier Dream Cycle stuff, start with Dreams in the Witch House; it's sort of a halfway mark between straight horror and the more surreal ones he did later on. Brown Jenkin is a creepy little fucker, I used to have actual nightmares about him.
Everyone bigs up At the Mountains of Madness, but apart from a memorable bit involving giant penguins I always thought it was overlong and not particularly scary. To be fair it's more concerned with world-building than actual horror, Lovecraft's Silmarillion in a way.
