DBZ was a good intro into Anime, but Cowboy Bebop is the standard for classic anime storytelling and considered a GOAT series.
It depends on the genre of anime you're into and if it's comparable to the manga (comic) of that same series. In Japan, you graduate to Japanese nuances, lifestyle, and story. I'll never forget watching Samurai Champloo in Japan as it came on TV around 1-2am late night as the language was too mature in Japanese (Kids didn't watch that, unlike Pokemon & Shin-chan). Full Metal Alchemist ranks high on the list. If you like stuff like Golden Boy, you'll love GTO (Great Teacher Onizuka).
My Picks:
If you like an interesting storyline, watch a series like My-HIME (first) then My-Otome (After watching My-HIME)
It's starts lighthearted and playful, then gets serious real quick as the story progresses!
Series about Japanese life and will have your emotions played with: Welcome to the NHK
Director: Anything by Satoshi Kon (Paranoia Agent is a Japanese classic)
My recent classic you can watch on Amazon Prime: Last Hero Inuyashiki ( the clash between the older and younger generations with story and action)