Gonna copy my comment from elsewhere bc it has four shows on netflix (hunter x hunter , evangelion, and avatar and FMA)
Besides miyazaki moviesnwhich are accessible enough everyone already knows about them and doesnt need an introduction, heres my brief guide to anime I've seen so far:
Mushishi --show about a travelling healer type pwrson in a world in which protozoan spirits (sorta) called mushi cause many diseases. Sorta slow, subdued, and in the sense of animism and connection to nature is similar to miyazaki. I havent finished it but still recommend.
Akira: not going to review in detail bc it's so well known but a classic for a reason one of the best anime and also best cyberpunk movies animated or not animated.
Ghost in the shell (1995): same review as akira word for word, even if different plot.
Hunter X Hunter: this has replaced evangelion as my favorite anime of all time, and is one of if not the favorite tv shows of all time of mine.
Impossible to easily or succinctly describe or categorize but I have to because I want to evangelize about it. it is a very adult show while having some of the charm and innocence of a children's one. Its brutal, but also eclectic and funny. It takes a world that essentially a lot like ours and adds vitalism and a system of quasi magical aura but doesnt use it as a deus ex machina or simple plot armor--despite the "magic" aura stuff, life and combat and everything is still incredibly difficult and complex for these people that also live in an alternate modernity, complete with cell phones , money, the mafia , etc. Its a story about a boy's quest to become a hunter"", which is a profession that doesnt have an equivalent in our world but is sorta like a combat proficient but also detective skills imbued Jack of all trades that can focus on any specialties and use their license to gain access from anything from ancient ruins to jobs bodyguarding mafia members , and so on and so on. The boys quest is mainly to find his father who abandoned him to work as. a hunter.
But there are many other, fairly baroque plots.
It's also a story about his friends, and they come equipped with their own potent and complex backstories and needs/visions. Every aspect of the world building is valuable. There is really almost nothing extraneous to the story.
It's the most nietzschean anime I've ever seen and maybe the most Nietzschean film or television I've seen. It has lots of stuff about the importance of will, loyalty, friendship, and courage, but also lots of relatively amoral esotericism. And ultimately it makes an implicit argument that the bonds of love and friendship matter more than any universal ethics.
It has child assassin nobility, mad bodhisattva warrior-monks, and way way more. It manages to be both cerebral and good at action sequences.
Every character has depth too.
It takes a few episodes or even more to become fully emotionally invested, at first it may come across as overly bizarre or cerebral but it quickly gets u totally hooked and once ur hooked u won't care how cerebral it is u will just watch even if confused and love every moment of it.
Another way to describe it is I think that it's a lot like moby dick, has that kind of maximalism and tragic /huge scope
Okay I used up a lot of my time writing about hunter x hunter and it's lateso for the other anime just gonna give number ratings
Neon genesis evangelion 4/5, overall, but get the subbed version by torrenting , the Netflix dub sucks
The End of evangelion (4.5 or 5/5)
Fullmetal alchemist (3.5/5) I'm harsh, this is a pretty damn good show but 3.5 from my harsh perspective means damngood but with imperfections. (HxH would be a 5 tho )
Attack on titan (1.5/5) really not much to say, interestj f concept and maybe the manga is good, terrible execution.
Cowboy bebop (havent finished yet but so far a solid 3.5-4/5)
Avatar the last airbender (some may not count it as anime but I think it basically is american anime with similar themes as a lot of anime. So 4/5)
The legend of korra (atla sequel) 3.5/5