The One Where Monica Gets a Roommate - S1-E1
Other mistake: When Rachel arrives at Central Perk, Monica introduces her to everybody, including to Chandler, as if they don't know each other. In later episodes, we learn that Chandler met Monica and Rachel a couple of times when they were in high school (including when Monica cut his toe off). Chandler even kissed Rachel once, and since Chandler always says that he hasn't kissed many women, he should have perfectly remembered Rachel, and Monica should have remembered that the two of them knew each other. (00:04:40)
Suggested correction: Chandler and Rachel never really knew each other well. Chandler will have been in her mind as that guy that her best friend's brother hangs out with, and Chandler will only know her as the friend of his friend's sister. I doubt they'd remember each other as the kiss they shared was when she was drunk, and they didn't see each other for 6 years while Rachel lived a richer and dependent on her father type life.
The One Where Monica Gets a Roommate - S1-E1
Other mistake: Ross is a paleontologist but is seen working with the display on early humans which would be more of a job for an anthropologist.
The One Where Monica Gets a Roommate - S1-E1
Other mistake: At the end when Ross and Rachel split an Oreo, you can see that Rachel is holding 2 parts and Ross somehow still has one. (00:20:38)