i want this to sound smooth

At first I didn’t believe that I could relate to my classmates in any way; we were all unique in terms of backgrounds and social classes.

2 answers

It's all in 1st person, but see if you like this any better:

At first I didn’t believe that I could relate to my classmates in any way: Each of us is unique in terms of backgrounds and social classes.
You could substitute "because" for the semi-colon.