In no particular order.
The Seeker: his quest, should he accept it, is to find the objects and save the world.
LOTR: Fellowship of the Ring: The Quest? Get rid of the ring!
Monty Python and the Holy Grail: self explanatory.
Raiders of the Lost Ark: To find the Ark of the Covenant.
Back to the Future: to return home.
Wizard of Oz: Sadly, just to get home.
Stardust: to capture the star to get the girl
Thoroughly Modern Millie: (Yes, I know I'm stretching it) Her quest? To find a rich husband. :-}
How to Marry a Millionaire: See the above.
Count of Monte Cristo: Yes, a quest for revenge.
The Gnome-Mobile: to find the little people!
MirrorMask: she must find the mask to save the fantasy world and her mother.
Willow: to destroy the evil queen!
So...what's your favorite quest movie?