I love haunting/possession thrillers like The Conjuring, The Exorcist (the original one, not the new one) and Insidious. Does anyone have any suggestions for more good haunting movies to watch?

Alien.
Silence of the Lambs.
Manhunter (original version).

torgo will be along shortly to help you. in the meantime here's a picture of a bunny with a pancake on it's head

Insidious had me sleeping with the lights on for a week! The Paranormal Activity movies are good too. When I get too scared I cheat and read the Wikipedia plot summary!

Night Of The Demon aka Curse Of The Demon (1957)
The Uninvited (1944)
The Haunting (1963)
The Innocents (1961)
The Shining (1980)
The Changeling (1980)
The Others (2001)
Session 9 (2001)
The Woman In Black (1989)
The Forgotten One (1989)
The Legend Of Hell House (1973)
Ju-On (2000)
Ju-On: The Grudge (2002)
The Entity (1982)
Let's Scare Jessica To Death (1971)
Burnt Offerings (1976)
Full Circle aka The Haunting Of Julia (1977)
Lovely Molly (2011)
Possession (1981)
The Hearse (1980)
The Nesting (1981)
The Innkeepers (2011)
The Orphanage (2007)
The Devil's Backbone (2001)
The Pact (2012)
House Hunting (2013)
Absentia (2011)
Shock (1977)
The Exorcist III (1990)
Shutter (2004)
The Sentinel (1977)
Audrey Rose (1977)
Wind Chill (2007)
Dead Birds (2004)
Lake Mungo (2008)

wow! Thanks! That's a huge list! I've seen about 10 of them

I've actually never seen silence of the lambs! Is it as good as they say?

It's excellent. As is Manhunter which is based on Harris' other book Red Dragon.