ANOTHER EDIT: Fixed a misread kana - it's "Chotto shobo kureteru no?" and not "shobu," which makes it "You're a bit disheartened, aren't you?" instead of the "It was a bit difficult, wasn’t it?" I got from that line before.
EDIT: Added the name of Joy's voice actor; fixed the early written text; looked up the kanji in "kocchimuite." Right, that's all I'll fix; the rest can be derived from the context.
All right, here we go. I don’t have much practice reading kanji (the more complicated Japanese letters), so please excuse the untranslated stuff. I have a large kanji encyclopedia, but looking them up takes forever, so for now, you get this “raw” and I’ll edit this post later.
First screen reads „Pikusaa (something) animeeshyon.“ Can’t read most of the kanji, but with the 20 and the kanji for “year” in there, it’s probably “Pixar’s 20th anniversary” or something.
Takeuchi Yūko (the first woman, Joy’s or “Yorokobi’s” voice actor):
I think the most important feeling I want to feel is happiness. So of course, laughing is the thing I like to do most.
(Ichiban daiji ni shitai kanji wa yorokobi da to omoimasu. Yappari waratteru jibun ga ichiban suki desu.)
Ōtake Shinobu (the second woman, Sadness’s or „Kanashimi’s“ voice actor):
If there’s someone who has/makes(?) sad memories, [you should] mourn together. That’s why I think humans are great/beautiful/lovely/what-have-you.
(Kanashii omoi wo shiteru hito ga ittara, issho ni kanashinde ageru. Dakara ningen tte suteki da na tte omoimasu.)
This is your story
(Kore wa, anata no monogatari)
Then, the song (I’m fairly certain about this part, at least, since I can hear and read the words):
All right, come here / hear what I have to say
(Saa koko he kite / hanashi kikasete)
What was today like? [Literally “what kind of day was today?”]
(Kyou wa donna hi datta?)
You're a bit disheartened, aren't you?
(Chotto shobo kureteru no?)
I know both sadness and anger well
(Kanashimi mo ikari mo ippai shitte)
It’s fun but also painful / Eat more and more [??! It says so… oO]
(Tanoshii mo tsurai mo / dondon tabete)
Become a big girl! [Literally “become big,” but “ooki” can be used for how tall one is, and with the eating above… yeah.]
Riley, Riley, look over here/this way
(Rairii Rairii kocchimuite)
Riley, Riley, wipe off your tears and raise your head [literally “face,” but that sounds weird to me]
(Rairii Rairii namida fuitara kao agete)
The fact that I have seen you change [with connotations of „grow“] each and every day
(Mainichi kawaru anata mite iru koto ga)
Is the greatest bliss
(Ichiban no shiawase)
Voiceover, probably from Joy/Yorokobi:
Believe in it, Riley, tomorrow will surely turn out to be a good day
(Shinjite Rairii, ashita wa kitto ii hi ni naru wa)
Inside Head [which is the Japanese title for the movie]
[something I can't read] ♪ Beloved Riley – DREAMS COME TRUE [which are the title and artist of the song]
((something something) ♪ Itoshii no Rairii / DREAMS COME TRUE)
Goodnight, Riley / Right now…
(Oyasumi Rairii / ima wa…)
Can’t read the kanji, but 7/18 is probably the date the movie will come to Japanese cinemas.
That's it! Hope someone finds this useful!