NGE straddles a few different genres. Broadly speaking it's a sci-fi drama. Narrowing it down, it's "giant robots vs monsters" and a "coming of age" story. The creator, Hideaki Anno, made a thing of giving conventions the finger, by subverting both these genres, giving audiences an ending they hated (and yet one which the entire series was carefully building up to), and then responding to their hatred with a movie that did its best to rape your face.
Oh, you thought I was just being figurative?
Subverting Giant Robots vs Monsters
The Evas first appear to be humanoid machines. They have pilots, a battery life and everything. But, by the end of the first battle, we learn that there's a lot more emphasis on the humanoid part of that equation, and that their machine parts are only on the outside.
The first thing we learn about the monsters is that they're angels. We're off to a good start here. We soon learn that the angels have an extremely close DNA match to humans (99.98% IIRC), that the Evas are made from one of the angels, and that the Evas have a human will inside them.
So, instead of big machines fight big monsters, we have mankind making slaves of giant humans-at-heart made from an angel in order to stop the angels. If any of that sounds like a recipe for disaster, good: now you're getting it.
Subverting the Coming of Age Story
The central character of Evangelion is Shinji Ikari, a deeply insecure, broken, psychologically frail, emotionally weak teenage boy. Everything in the story is about Shinji. Everything. All the stuff that's obviously, directly about him; all the issues of everyone around him; the battles which we all initially think are about cool action, explosions and thrashing Tokyo-3; even the superficially silly comic relief stuff: all of it is there for the purpose of taking the boy and making a man.
Doesn't sound very subversive yet. Actually,t hat sounds like the pinnacle of the coming of age genre. Here's where the subversion comes in: the series is essentially a massive 3-act play. The first act introduces Shinji and his fantasies. The second act uses every character and scenario to build him up to the point that he's ready to make the decision to be a man. Then in the final act, he will lose everything that's been holding him up enabling him to be a man. This is where the subversion comes in. We expect that the decision to man up means that he's going to conquer his fears and master his life from here on out. Nope. Everything that's brought him to the point of manning up is going to be ripped right out from underneath him. We're used to the hero turning down the help of his friends because "This is something I have to do myself." As for Shinji, he's desperately begging for anyone and everyone to help him, but against his will this is something he has to do himself. If he's going to be a man, then he's going to do it all on his own. And remember his starting point: he's a deeply insecure, broken, psychologically frail, emotionally weak teenage boy. So this is serious make or break stuff. Sink or swim. Master yourself or lose yourself.
Make or break? Break it is!
About Act 1
I'm going to draw the line and say that episodes 1-7 are Act 1. Here are some key features of the first act.
Shinji's relationship with Rei: Rei Ayanami is one of the first people we meet, after Shinji. Her initial appearance is for a brief moment, standing calmly in the middle of the road, and then disappearing in a very Batman-esque style ("Oh, I see you're blinking. This seems like an appropriate moment for me to destroy all evidence that I was ever here."). This moment makes little sense until you see the movie, because it is actually inconsistent with what is revealed about her 5 minutes later: that she's severely injured, sleeping/unconscious (until woken up), covered in bandages and barely able to sit up without yelping in pain.
There's a lot of sexual humour in Act 1, including a certain scene in Rei's apartment which, when I was Rei and Shinji's age, was one of my all-time favourite scenes of the series. Alongside all the awkward sexualisation and sexual humour, Rei symbolically plays the role of Shinji's fantasy of femalekind. She's physically attractive in a strictly technical sense (ie she's structurally attractive), she's profoundly obedient, she's the only girl (that we know of at this time) going through what Shinji's going through, and she's frequently the damsel in distress (this last point is what initially motivates Shinji to pilot Eva). It all seems like they're meant to be together. But, as the fantasy rather than the reality (symbolically speaking), things don't get to play out that way. Rei is emotionally distant, and, from her end, emotional engagement with Shinji only happens when his knight-in-shining-armour-ness reminds her of Shinji's father. If that sounds creepy, don't worry. It gets worse. More on that later.
Gendo Ikari saves Rei from being boiled in her entry plug
Shinji Ikari saves Rei from being boiled in her entry plug
Shinji's relationship with Kensuke and Toji: Kensuke Aida and Toji Suzuhara are two of Shinji's classmates, and the only people we'll really see become Shinji's friends throughout the bulk of the story. Kensuke, like Shinji, has no mother. Kensuke in some sort of idealised form of Shinji: we never see him showing any signs of insecurity or emotional weakness, and while Shinji's the frontman of a war he doesn't want to be a part of, Kensuke is a war-enthusiast, lamenting the fact that he doesn't get to be a part of the war.
Toji's first impressions of Shinji aren't all the flattering, unless you consider a heavy fist or two in the face a nice way to say hello. Shinji's first Eva battle has left Toji's sister severely wounded, so when Toji finds out that Shinji was piloting Eva Unit 01, Toji's not impressed. Later, after Toji and Kensuke's lives are directly saved by Shinji in the Eva, Toji insists that Shinji punch him back. Shinji does it, but immediately regrets it, insisting that he's the one who deserves to be punched, not Toji. Symbolically, this conflict and attitude has huge ramifications in Acts 2 and 3, playing out on a much larger scale.
Shinji's daddy issues: We don't learn much about Gendo Ikari in the first act, even though he's constantly around giving orders. We get to see that he's gruff and unfriendly right off the bat, that he has a cold employer/employee relationship with Shinji (which has been inactive for 3 years before the story starts), and that Shinji hates him. In episodes 5 and 6, we see Gendo caring about Rei, and Rei, who normally expresses very little of anything, lighting up when he speaks to her. Shinji freaks out a bit at the sight of this. It isn't clear to me whether he's jealous of Rei attaining his father's affection, or of his father attaining Rei's affection, so just to be safe I'm going to say it's both.
Shinji's mamma issues: We don't learn much about Shinji's mum at this point in the story. In fact, I'm not even sure if she's mentioned. But she has a great substitute in Misato Katsuragi. And by great, I mean Shinji's quick appraisal upon seeing the inside of her apartment is: "Ice....snacks....and fifty gallons of beer! What kind of life does she lead?!" Misato functions as Shinji's guardian throughout the series, which is made creepy by a whole lot of awkward sexual humour between the two early on, which reverberates later on. The sexual tension seems to be dismantled a lot through Shinji being unimpressed with Misato's sloppy housekeeping.
The sexual stuff goes deeper than awkward humour, though. The Third Angel (which is the first enemy of the series) is symbolic of Shinji's relationship with womankind. In particular, it's symbolic of when that relationship is damaged. Eva Unit 01 rips out one of the angel's ribs and uses that rib as a weapon, referencing woman being made of man's rib in the Bible. The angel, realising that it's going to die, decides that if it has to die, it may as well take down the Eva and everything else with it, indicating the messy crap that tends to happen during a break-up.
In turn, the Fourth Angel represents Shinji's estranged relationship with malekind. I almost don't want to write this, because I was able to go so long without seeing it, and once it's been seen, it can't be unseen. For years I was able to look at the Fourth Angel and just see a giant, buggy worm. And I like it that way. I want it to go back to that way. But again, what has been seen cannot be unseen. The Fourth Angel is a giant penis.
It's even got the...and the....and the way it...
Well, I'm overdue for sleep now. I'll continue this with Acts 2 and 3 once I've had some more sleep, and actually have the time to write. I'd say goodnight, but after seeing a penis like that...