LOTR for fantasy for sure
SciFi: That's a real tough one. When I was younger certain Old SciFi movies stood out, though I don't recall their names.
1) "Year 0"(I believe) a story of a LA family on vacation when a Nuclear Holocaust occurs and their fight for survival. Certainly not the best movie of that genre, but it had a creepy reality showing mass panic and people's decent into barbarism. If I were to watch it today I'd probably not like it, but it has been a long time.
2) Another creepy movie from the 60's which I have no recollection of the title was about a disease that kills off most humans, which had a habit of returning as zombies and killing their family members, and 1 mans' struggle to survive(he later finds a few more uninfected Humans). Probably wouldn't like this one either, but thought it was way cool way back when.
3) thee are a ton of Japanese and US space movies from the same era (50s-60s) that are also fairly good on some levels at least
4) My favorites though are more recent: Mad Max Road Warrior, Blade Runner, Matrix trilogy, first 2 Aliien movies, 2001 A Space Odyssey, first 3(made) Star Wars films, BattleStar Galactica(original, though remake isn't bad), aaaargh too many to list or choose from