At first I didn’t believe that I could relate to my classmates in any way; they all came from different backgrounds and social classes, but as time progressed we came to know each other as family rather than peers.
The sentence is grammatically correct. I would put a comma after 'first' and a comma after progressed.
NOTE - I think the sentence is a little long. Were it I, I would put a period after classes, and start the new sentence However, as time progressed, we.....