The best series for you to check out really depends on what you want to see. If you really want to check out the classics, Cowboy Bebop is probably the most widely acclaimed series. Neon Genesis Evangelion is considered a classic too. If you want something more recent but smart and dramatic, check out Death Note or Steins;Gate. On the other hand if you want to just see people punch each other, you could check out Naruto or One Piece.
My personal favorite is Tengen Toppa Gurren Lagann, but it's probably too over the top if you're new to anime.
I like one piece, naruto, hajime no ippo, shokugeki no soma, and many others but my favorite definitely is zetsuen no tempest, also you should watch death note and Tokyo ghoul if you haven’t seen it yet.
And series.... I don’t know, maybe game of thrones and breakin bad (I’m not a series guy)
FLCL without a doubt. It says so much in such a confusing way and it's truly an experience worth every second of your time. Some people don't like the sequels, but I like them too. But definitely the original