While there is an excellent walkthrough already posted by Fernando Esra, you'd have to do 17 playthroughs to collect all 21 endings with it, and this amount can be greatly reduced by combining routes together and saving/reloading at the appropriate places. I took all the routes from that guide and combined them into a tree to see how far different routes intersected, and used that to form a route utilizing saves to reduce the amount of duplicate text you have to skip through. This can most likely be optimized significantly more, as there are more than one way to get many endings, but as I don't know the actual criteria, I just treated the known routes as THE routes and stuck with them, and it still should be a significant improvement.

This route uses 5 save slots. Whenever you see save #, save in that slot before choosing the next choice.

Refactored Dec 7. I didn't realize some of these had prerequesites originally. New route should match them in order.

Dec 9 fixed missing choices in 6,7,8,11,12

Dec 14 removed non-existant choice from 18. I have now completed a playthrough through this and am confident it is all correct.

Split into 4 parts. Part 1 starts off with visiting Amu, part 2 starts with visiting Noelle, part 3 resumes part 1, and part 4 resumes part 2. Switching back and forth is needed to get the prerequesites. Endings 19 and 20 could have been attached to either part 2 or 4, but there's no difference in time, so I attached them to part 4 so the endings are more in order.

visit amu
cool yourself down
you'd look cute as a maid
buy her an ice cream
save 1
go home with Amu
save 2
approach Noelle
thank you for being so kind
go and visit Noelle
why are you moving?
tell her you're here
save 3
I like you both the same
ending 1
load 3
I like you more than Amu
ending 2
load 2
just go home
doesn't matter
doesn't matter
doesn't matter
save 2
go and talk to Noelle
doesn't matter
ending 4
load 2
it's already too late. just stay here
save 2
go with her
ending 13
load 2
stay here
save 2
tell a lie
ending 14
tell the truth/I'm sorry if I hurt you/for lying to you
ending 15

part 2(3, 5, 6, 7, 8, 9, 10, 11, 12):
visit Noelle
save 2
don't buy her an ice cream
ask how Noelle is
I can't help but worry
save 3
I'll just leave it
go home by myself
approach Noelle
save 4
thank you for being so kind
don't pry any further
visit Noelle
why are you moving?
tell her you're here
invite Noelle to go with you
doesn't matter
ending 5
load 4
you really mean a lot to me
where are you moving?
go and visit Noelle
where are you moving?
wait a little longer
hug her
don't curse me Amu
rest doesn't matter
ending 10
load 3
I do have an idea
approach Noelle
you really mean a lot to me
where are you moving?
tell Saki
visit Amu
doesn't matter
it sounds tough. hang in there
doesn't matter
call Saki
invite Noelle to go out with you
doesn't matter
doesn't that mean you're lonely?
ask about the painting
ending 9
load 2
buy her an ice cream
save 2
don't want to hear about it
go home by myself
keep going down the same road
approach Noelle
thank you for being so kind
don't pry any further
visit Noelle
why are you moving?
tell her you're here
invite Noelle to go with you
doesn't matter
doesn't that mean you try hard?
ending 6
load 2
ask how Noelle is
save 2
I can't help but worry
I'll just leave it
go home with Amu
approach Noelle
thank you for being so kind
do you want to move?
tell her
visit Noelle
why are you moving?
tell her you're here
I like you more than Amu
ending 3
load 2
maybe I shouldn't worry too much
save 2
I'll just leave it
go home by myself
save 3
keep going down the same road
approach Noelle
you really mean a lot to me
where are you moving?
save 4
visit Amu
I'm sorry it was a stupid joke
save 5
it sounds tough, hang in there
doesn't matter
invite Noelle to go with you
doesn't matter
doesn't that mean you're lonely?
ask about the painting
ending 7
load 5
I think you're amazing
doesn't matter
invite Noelle to go with you
doesn't matter
doesn't that mean you're lonely?
ask about the painting
ending 8
load 4
visit Noelle
where are you moving?
wait a little longer
hug her
don't curse me Amu
if you want, it can be a date
buy the doll
save 4
the person I care about the most
ending 11
load 4
the person I like the most
ending 12

load 1
go home by yourself
go down the sunflower road
keep talking to her
save 1
these things happen
just go home
doesn't matter
I know Eri is annoying, but
ending 16
load 1
I'm sorry to hear that
save 1
just go home
doesn't matter
I don't think Eri is a bad person though
accept the offer
go with Tsuzumi
ending 17
load 1
see Noelle
thank you for being so kind
doesn't matter
I don't think Eri is a bad person though
accept the offer
stay with Eri
I suppose we are friends
ending 18

load 3
go down to the sunflower road
keep talking to her
I'm sorry to hear that
talk to Noelle
you mean a lot to me too
do you know where you're moving?
choice doesn't matter
I don't think Eri is a bad person though
accept the offer
stay with Eri
I suppose we are friends
save 3
don't ask her
ending 19
load 3
ask her
ending 20
load 2
I do have an idea
ask her about it
no more choices matter
ending 21

Thanks to Fernando Esra for making the guide this is based on, and Ebihime for making the guide that guide is based on.

1: visit amu(1, 2, 4, 13, 14, 15, 16, 17, 18)
_2: cool yourself down(1, 2, 4, 13, 14, 15, 16, 17, 18)
_3: you'd look cute as a maid(1, 2, 4, 13, 14, 15, 16, 17, 18)
_4: buy her an ice cream(1, 2, 4, 13, 14, 15, 16, 17, 18)
_5: go home with Amu(1, 2, 4, 13, 14, 15)
__6: approach Noelle(1, 2)
___7: thank you for being so kind(1, 2)
___8: go and visit Noelle(1, 2)
___9: why are you moving?(1, 2)
___10: tell her you're here(1, 2)
___11: I like you both the same(1) - ending 1
___11: I like you more than Amu(2) - ending 2
__6: just go home(4, 13, 14, 15)
___7: doesn't matter(4, 13, 14, 15)
___8: doesn't matter(4, 13, 14, 15)
___9: doesn't matter(4, 13, 14, 15)
___10: go and talk to Noelle(4)
____11: doesn't matter(4) - ending 4
___10: it's already too late. just stay here(13, 14, 15)
____11: go with her(13) - ending 13
____11: stay here(14, 15)
_____12: tell a lie(14) - ending 14
_____12: tell the truth/I'm sorry if I hurt you/for lying to you(15) - ending 15
_5: go home by yourself(16, 17, 18)
__6: go down the sunflower road(16, 17, 18)
__7: keep talking to her(16, 17, 18)
__8: these things happen(16)
___9 just go home(16)
___10: doesn't matter(16)
___11: I know Eri is annoying, but(16) - ending 16
__8: I'm sorry to hear that(17, 18)
___9: just go home(17)
____10: doesn't matter(17)
____11: I don't think Eri is a bad person though(17)
____12: accept the offer(17)
____13: go with Tsuzumi(17) - ending 17
___9: see Noelle(18)
____10: thank you for being so kind(18)
____11: doesn't matter(18)
____12: I don't think Eri is a bad person though(18)
____13: accept the offer(18)
____14: stay with Eri(18)
____15: I suppose we are friends(18) - ending 18
1: visit noelle(3, 5, 6, 7, 8, 9, 10, 11, 12, 19, 20, 21)
_2: buy her an ice cream(3, 6, 7, 8, 11, 12, 19, 20, 21)
__3: ask how Noelle is(3, 7, 8, 11, 12, 19, 20, 21)
___4: I can't help but worry(3)
____5: I'll just leave it(3)
____6: go home with Amu(3)
____7: approach Noelle(3)
____8: thank you for being so kind(3)
____9: do you want to move?(3)
____10: tell her(3)
____11: visit Noelle(3)
____12: why are you moving?(3)
____13: tell her you're here(3)
____14: I like you more than Amu(3) - ending 3
___4: maybe I shouldn't worry too much(7, 8, 11, 12, 19, 20, 21)
____5: I'll just leave it(7, 8, 11, 12, 19, 20)
_____6: go home by myself(7, 8, 11, 12, 19, 20)
_____7: keep going down the same road(7, 8, 11, 12)
______8: approach Noelle(7, 8, 11, 12)
______9: you really mean a lot to me(7, 8, 11, 12)
______10: where are you moving?(7, 8, 11, 12)
_______11: visit Amu(7, 8)
________12: I'm sorry it was a stupid joke(7, 8)
________13: it sounds tough, hang in there(7)
_________14: doesn't matter(7)
_________15: invite Noelle to go with you(7)
_________16: doesn't matter(7)
_________17: doesn't that mean you're lonely?(7)
_________18: ask about the painting(7) - ending 7
________13: I think you're amazing(8)
_________14: doesn't matter(8)
_________15: invite Noelle to go with you(8)
_________16: doesn't matter(8)
_________17: doesn't that mean you're lonely?(8)
_________18: ask about the painting(8) - ending 8
_______11: visit Noelle(11, 12)
________12: where are you moving?(11, 12)
________13: wait a little longer(11, 12)
________14: hug her(11, 12)
________15: don't curse me Amu(11, 12)
________16: if you want, it can be a date(11, 12)
________17: buy the doll(11, 12)
________18: the person I care about the most(11) - ending 11
________18: the person I like the most(12) - ending 12
______10: visit Amu(8)
_______11: I'm sorry, it was a stupid joke(8)
_______
_____7: go down the sunflower road(19, 20)
______8: keep talking to her(19, 20)
______9: I'm sorry to hear that(19, 20)
______10: talk to Noelle(19, 20)
______11: you mean a lot to me too(19, 20)
______12: do you know where you're moving?(19, 20)
______13: choice doesn't matter(19, 20)
______14: I don't think Eri is a bad person though(19, 20)
______15: accept the offer(19, 20)
______16: stay with Eri(19, 20)
______17: I suppose we are friends(19, 20)
______18: don't ask her(19) - ending 19
______18: ask her(20) - ending 20
____5: I do have an idea(21)
_____6: ask her about it(21)
_____7: no more choices matter(21) - ending 21
__3: don't want to hear about it(6)
___4: go home by myself(6)
___5: keep going down the same road(6)
___6: approach Noelle(6)
___7: thank you for being so kind(6)
___8: don't pry any further(6)
___9: visit Noelle(6)
___10: why are you moving?(6)
___11: tell her you're here(6)
___12: invite Noelle to go with you(6)
___13: doesn't matter(6)
___14: doesn't that mean you try hard?(6) - ending 6
_2: don't buy her an ice cream(5, 9, 10)
__3: ask how Noelle is(5, 9, 10)
__4: I can't help but worry(5, 9, 10)
__5: I'll just leave it(5, 10)
___6: go home by myself(5, 10)
___7: approach Noelle(5, 10)
___8: thank you for being so kind(5)
____9: don't pry any further(5)
____10: visit Noelle(5)
____11: why are you moving?(5)
____12: tell her you're here(5)
____13: invite Noelle to go with you(5)
____14: doesn't matter(5) - ending 5
___8: you really mean a lot to me(10)
____9: where are you moving?(10)
____10: go and visit Noelle(10)
____11: where are you moving?(10)
____12: wait a little longer(10)
____13: hug her(10)
____14: don't curse me Amu(10)
____15: rest doesn't matter(10) - ending 10
__5: I do have an idea(9)
___6: approach Noelle(9)
___7: you really mean a lot to me(9)
___8: where are you moving?(9)
___9: tell Saki(9)
___10: visit Amu(9)
___11: doesn't matter(9)
___12: it sounds tough. hang in there(9)
___13: doesn't matter(9)
___14: call Saki(9)
___15: invite Noelle to go out with you(9)
___16: doesn't matter(9)
___17: doesn't that mean you're lonely?(9)
___18: ask about the painting(9) - ending 9