There's nothing better for breaking someone in to horror movies IMO than to show them short horror films, which are freely available online. It's sort of like getting samples of ice cream at the shop to see which one you like, without the inconsideration of holding up a large line to try the lot. I've tried to rank them from least to most terrifying. Anyway, here are some that are all great:
The Black Hole
Exit (with Julian Glover)
The Jigsaw (about a puzzle, not the movie series)
2 AM - The Smiling Man
He Dies at the End
The Thing in the Apartment
There are so many great short films online that you could have a marathon that lasts a solid weekend. Try the links to others you'll find on YouTube, but note that some can be very scary and/or gory, while many others are not very good at all. If it is stamped with "award winning" or the like, then that is usually a good bet.