Intro (verse) 0:00-0:04 A minor
Verse 1 0:04-0:33 A minor (with a touch of C)
Verse 2 0:33-0:59 A minor (with a touch of C)
Middle 8 0:59-1:15 A major
Verse 3 1:15-1:41 A minor (with a touch of C)
Middle 8 1:41-1:57 A major
Verse 4 1:57-2:23 A minor (with a touch of C)
Coda 2:23-2:34 A minor
Comments: "Things We Said Today" employs a very similar verse structure as  "A Hard Day's Night". The verse may be split into four parts:
You say you will love me if I have to go.
You'll be thinking of me, somehow I will know.
Someday when I'm lonely, wishing you weren't so far away,
Then I will remember things we said today.
The only difference between the first and second parts is the lyrics. But then with the third segment, the chords change and vocal harmony is added before reverting to the characteristics of the first two segments for the final part.
Like  "Any Time At All", "Things We Said Today" plays between A major and A minor. It is no coincidence that these two were recorded consecutively.