Fullmetal Alchemist Brotherhood is really excellent and I think they just added it back to Netflix. I'd definitely recommend it, it has a good mix of serious action with some lighter elements, and is really well done. Both eminently watchable and invites some reflection. There's also the original FMA which diverges from the story of Brotherhood (which follows the manga), due to at the time overtaking the manga. Worth a watch on its own merits to see where it goes and develops characters differently (a couple have very different arcs which are really interesting in their own).
Cowboy Bebop is also absolutely fantastic and really a seminal anime I think. I believe it is on Netflix (the anime that is of course, not the live action version which I hear wasn't great). Really thought provoking and layered, a lot of depth for analysis. Definitely a "changes your brain chemistry" type of media work.
Ghost in the Shell: Stand Alone Complex (and 2nd Gig) is great but I don't think it's on any streaming services (the original film might be i.e. the anime film not the live action one). Some really interesting concepts in the worldbuilding of it all as well as good character writing and chemistry between the characters I think. Worth watching just for the tachikoma at least!
Dungeon Meshi (Delicious in Dungeon) is pretty fun and cute, has some more serious bits but overall relatively light-hearted, and is on Netflix.
I've heard Neon Genesis Evangelion is very good and something that one "ought to watch" if into anime (whatever importance you put on that anyway), I've never got around to actually watching but I'm pretty sure I've seen it on Netflix.
Can't really comment on much in the slice of life or magical girl genres as I haven't watched much in those (except Sailor Moon in the latter genre which I watched as a kid and loved at the time). Likewise for the mecha genre (which I know of NGE but haven't really watched it or anything else, like Gundam, to comment).
Also try checking out local CEX stores or similar as you can probably find a lot of preowned anime series on disc for relatively cheap there.