Here are some must sees.
Alien. The original one.
The Apocalypse Trilogy by Carpenter: The Thing, 1982 version (don't you dare watch the new version); Prince of Darkness, 1987; and In the Mouth of Madness, 1994.
Let the Right One In (NOT THE AMERICAN VERSION, FOR HADES SAKE!)
The Ring (This is a rare case of an American remake being good.)
No One Lives (One of the only slasher movies I enjoyed, and the only one I recommend.)
Ju-on - The Grudge, subtitled, and only the Japanese version. While the American remake was good due to using the same director and actress for Sadako, there are nuances in the original are missed.
Infection (Japanese, Subtitled). You will never go to a Japanese hospital again.
Don't Be Afraid of the Dark. Tooth Faeries. Yeah.
An American Werewolf in London. Don't bother with the sequel.
Slither, for humor with your horror.
Evil Dead 2 (Note the number! It's important!), for horror with your humor.
Evil Dead, 2013 Yikes. Just yikes.
Once you've seen these, you'll have a good grasp of the genre. Go forth, be scared.