Not sure... In the Fellowship (the first), there's Moria and the journey to Rivendell (while pursued by the Nazgul), which can be scary if you're young or sensitive or something like that. (Since I had read the books first, nothing was scary really, so I'm not sure.) The third one is probably the least scary, but there's Shelob (a giant spider), which can be "gory" depending on your definition of it. The second one is therefore probably the best one (but it's the one with the biggest battle, if you find such things scary.)
Why?