See Teaching Plato.
Nietzsche writes in the second aphorism from the section What I Owe to the Ancients of his work Twilight of the Idols, or, How to Philosophize with a Hammer (1888):
(trans. Anthony Ludovici):
I am not indebted to the Greeks for anything like such strong impressions; and, to speak frankly, they cannot be to us what the Romans are. One cannot learn from the Greeks— their style is too strange, it is also too fluid, to be imperative or to have the effect of a classic. Who would ever have learnt writing from a Greek! Who would ever have learned it without the Romans!… Do not let anyone suggest Plato to me. In regard to Plato I am a thorough sceptic, and have never been able to agree to the admiration of Plato the artist, which is traditional among scholars. And after all, in this matter, the most refined judges of taste in antiquity are on my side. In my opinion Plato bundles all the forms of style pell-mell together, in this respect he is one of the first decadents of style: he has something similar on his conscience to that which the Cynics had who invented the satura Menippea. For the Platonic dialogue —this revoltingly self-complacent and childish kind of dialectics— to exercise any charm over you, you must never have read any good French authors —Fontenelle for instance. Plato is boring. In reality my distrust of Plato is fundamental. I find him so very much astray from all the deepest instincts of the Hellenes, so steeped in moral prejudices, so pre-existently Christian— the concept ‘good’ is already the highest value with him— that rather than use any other expression I would prefer to designate the whole phenomenon Plato with the hard word ‘superior bunkum’, or, if you would like it better, ‘idealism’. Humanity has had to pay dearly for this Athenian having gone to school among the Egyptians (—or among the Jews in Egypt?…) In the great fatality of Christianity, Plato is that double-faced fascination called the “ideal”, which made it possible for the more noble natures of antiquity to misunderstand themselves and to tread the bridge which led to the ‘cross’. And what an amount of Plato is still to be found in the concept ‘church’, and in the construction, the system and the practice of the church!—My recreation, my predilection, my cure, after all Platonism, has always been Thucydides. Thucydides and perhaps Machiavelli’s principe are most closely related to me owing to the absolute determination which they show of refusing to deceive themselves and of seeing reason in reality,—not in ‘rationality’, and still less in ‘morality’.
There is no more radical cure than Thucydides for the lamentably rose-coloured idealisation of the Greeks which the ‘classically-cultured’ stripling bears with him into life, as a reward for his public school training. His writings must be carefully studied line by line, and his unuttered thoughts must be read as distinctly as what he actually says. There are few thinkers so rich in unuttered thoughts. In him the culture ‘of the Sophists’—that is to say, the culture of realism, receives its most perfect expression: this inestimable movement in the midst of the moral and idealistic knavery of the Socratic Schools which was then breaking out in all directions. Greek philosophy is the decadence of the Greek instinct: Thucydides is the great summing up, the final manifestation of that strong, severe positivism which lay in the instincts of the ancient Hellene. After all, it is courage in the face of reality that distinguishes such natures as Thucydides from Plato: Plato is a coward in the face of reality— consequently he takes refuge in the ideal: Thucydides is master of himself— consequently he is able to master life.
https://sententiaeantiquae.com/2019/03/23/plato-is-boring/. Werner Dannhauser has an article on Nietzsche and Plato, I think, that I should look at.
Nietzsche’s Genealogy of Morals beef with Christians, though is that Jew and Christians have slave morality, portraying victims, weak people, as being good and strong people as bad. Plato in no way elevates the humble and weak. There is something to write on there about Nietzsche confusing things. Nietzsche really has two distinct and not especially overlapping objections. First, he doesn’t like the elevation of humility and the ordinary person vis a vis the Noble Man, which Christianity does by saying that God is so far above everybody else that minor differences in human character and ability don’t matter. And, in the bad case, people elevate victims simply because they are victims, weak and inferior, out of envy of real virtue. That is a false charge against Christianity and Judaism, but a true charge against modern neomarxists, and a true charge against a lot of non-intellectuals too, who hate talent because they are so proud. Second, though, Nietzsche doesn’t like the elevation of the other-worldly over the earthly. That is his fight with Plato, and part of his fight with Socrates maybe. Nietzsche is not a materialist, but he wants people to be earthy (not just earthly, of course) and genuine, and not intellectual. And he thinks that people should live in the Present, not the Future (I don’t know what he thinks about the Past). A lot of this comes from Nietzsche’s Christian, German, Victorian, bourgeois upbringing, which he tries constantly to escape. And fails constantly. He never can get away from being a German Herr-Doktor-Professor and a moralist. And he knows it, I think. He is too smart not to. He knows he is not the Overman. And he blames Society for that.