Not counting Pokemon, DBZ, Ghibli movies etc... the earliest anime I watched was the first GITS movie, the first Eva movie, and Ergo Proxy. I had been browsing imageboards for years by then, but I always thought of anime as something pretentious, childish, and perverted because of the images that anons posted, so I only allowed myself to watch the more mainstreamy animes or the ones "mature" enough to be like Hollywood movies, but now I watch anime almost exclusively.
I really started watching anime when my bro introduced me to Attack on Titan and my sis introduced me to Fairy Tail, for Fairy Tail - I fell in love with the show's comedy, but then as I watched more shows I realized that it's the standard kind of comedy in anime, but for AoT - I hated Eren, but I thought it was pretty badass anime anyway and even now I still think it's a great anime despite how much more refined my tastes have become. I get that lots of anons consider it to be a "normalfag" anime for plebs, but if you can look past the normalfag reasons for liking it, there are deeper, real reasons for it being a genuine masterpiece. After that I started watching Kill la Kill, TTGL, Boku Dake Inai Machi, Gintama, and that's how I became a weeb.
>which series (current/past seasons) are meant to watch /late/ at night?
Fate Zero or anything related to Fate Stay/Night
Flip flappers (this one made me more depressed than I was but it's really good)
Pretty much any 00s anime does it for me. I like those feelings of nostalgia late at night, and the old art style makes it feel like time is broken.
Haven't watched these yet myself, but I'd watch them /late/ at night:
Black Rock Shooter
Serial Experiments Lain