Characters:
Rydia - RHIH-dee-ya ("rhih" sounding like the first vowel sound in rickshaw)
Celes - sell-LESS (like the first two syllables in celestial)
Cyan - SAI-yan
Sabin - SAH-bin
Gau - Gao/Gaw (pronounced it like Gao [sounds like gouge] when I was a kid, and now I pronounce it Gaw [like in gaudy])
Locke - Lock
Yuffie - YUFF-ee ("yuff" rhymes with buff)
Tifa - TEE-fuh
Aeris - Air-REEZ (pronounced just like the zodiac Aries)
Quistis - KWIH-stiss
Quina - KWEE-nuh. (I also pronounce it FAHT BAH-stud.)
Amarant - Ah-mer-ont (first syllable "ah" rhymes with first syllable of ostensive , and last syllable "ont" sounds like the first syllable of ontology)
Eiko - AY-koh
Ramza Beouvle - ROM-zuh BAY-oh-vule
Mustadio - Moo-STAH-dee-oh

Places:
Damcyan - DOM-ee-khan (last syllable sounds like "KHAAAAN!" from Star Trek, only not as sustained). I still pronounce this like I did when I was a kid. I believe I mentally switched the 'y' and 'c' characters.
Thamasa - THA-mah-suh
Narshe - NARSH (rhymes with marsh)
Kohlingen - KOH-leeng-in (middle syllable sounds like the first syllable in linger)
Mobliz - MOE-blizz
Tzen - Zen
Nibelheim - KNEE-bull-HIME
Dollet - Doll-LAY (pseudo-French pronunciation)
Esthar - ESS-tar
Oeilvert - AIL-vair (pseudo-French pronunciation)

Espers/Summons/GFs/Eidolons/long video sequences:
Ifrit - EYE-frit
Shiva - SHEE-vuh
Ramuh - RAH-moo (first syllable sounds like the Egyptian sun god)
Bahamut - as a kid: bah-HAH-moot; now: BAH-hah-mutt
Quetzalcoatl - KWEZ-all-coat-ull
Catoblepas - kat-TOE-bull-poss
Raiden - RAY-den

Bosses & Miscellaneous:
chocobo - CHUCK-oh-bow
cactaur - cack-TAR (first syllable sounds like first syllable in cackle)
Zeromus - ZAIR-oh-moose (first syllable sounds like air)
Necron - NECK-ron
Esuna - AY-soo-nuh (second syllable rhymes with soup)
ether - ETH-err (first syllable sounds like the name "Beth" but without the B)
genji - GEN-gee