You should probable memorise a lot of quotes for each principle character, as well as quotes for more minor characters, like Wolfshiem, Myrtle and Wilson, as well as quotes from minor characters about main characters, such as all the rumours about Gatsby, and what Catherine says about Myrtle and Tom hating their spouses, and so on...
It would be really great for your understanding of the text and the relationship between characters to memorise lots of quotes, especially ones that concern multiple characters, such as Jordan talking about Daisy and Gatsby to Nick.
Also maybe try to remember a variety of quotes that aren't just dialog between characters or descriptions. Take note of adjectives and phrases or literary devices Fitzgerald uses, such as the prominent water related words and colour symbolism. You can unpick these smaller quotes and lump them together as evidence of a certain semantic field or atmosphere that Fitzgerald tries to create.
At the very least I feel like you should remember at least 5-10 quotes for each character and a solid understanding of who each character is, their motivations and their role in the story and what they might represent, as well as the contextual value of this character, such as Daisy being a flapper and what that means for her character as a mother and a women, or Gatsby as a new money person who isn't accepted by the old money bourgeoisie. This would make your essays really convincing and accurate because you are familiar with the characters.
Also read the book again, maybe even multiple times, and keep a list of the important or notable quotes that you think would support a point or idea you're trying to make about a character or theme - think of quotes like evidence in a court case. If you want to win and you want the judge to believe and appreciate your claim, you will need a LOT of evidence that is reliable and strong that supports your points and make them solid.
You'll also need to know quotes for each theme in the book, important settings as well as important quotes that change or move the plot forward in an important way, such as the revelation that Gatsby believes he can reveal the past. Also it's crucial to know the story really well such as what goes on in each scene / chapter and what that means for the whole story, how does an event or revelation affect the story. Maybe do a detailed summary for each chapter and note down important quotes.
Hope this helps