So it starts as
I -IV[sup](b7)[/sup] - I - IV[sup](b7)[/sup] -
I - I[sub]/7[/sub] - vi - III - V[sup]7[/sup] - I - IV - vi - ii - [sup]#[/sup]iv[sup]7b5[/sup] - V - I
(the rest is left as an exercise in listening for you to finish....)
In this context the [sup]b7[/sup] means dominant 7 (i.e. a major chord with a minor 7th added) - it is in parentheses because it is played as a decoration note on the cello.
This seems to be in E[sup]b[/sup] so the progression ends up being:
E[sup]b[/sup] -A[sup]b(7)[/sup] - E[sup]b[/sup] -A[sup]b(7)[/sup] -
E[sup]b[/sup] - E[sup]b[/sup][sub]/D[/sub] - C min - G[sup]7[/sup] - B[sup]b7[/sup] - E[sup]b[/sup] - A[sup]b[/sup] - Cmin - Fmin - Am[sup]7b5[/sup] - B[sup]b[/sup] - E[sup]b[/sup]
Most telling of all is that I did the transcription in the Roman numeral form (without reference to an instrument) in one go.....
This backs what kebeats said, get your ear-trained and this will be nothing but a game to you..... go to
teoria.com and start using their ear training tools to boost your skills