Claymore is a decent combat oriented anime, but they kinda messed up the ending (changed it from manga), and there is a chance for a remake, so I'd wait for that.
Cowboy beebop is an all time classic soft sci-fi show, episodes work as standalone adventures, but there is overarching story.
Ghost in the shell (1995) and the sequel (Innocence, 2004) are both good though provoking sci-fi movies.
Berserk (1997 dark fantasy) is another classic and works well as an introduction to what some people consider the best manga of all time.
Chronicles of Lodoss War (1990-91) is an oldie but goldie light-ish combat oriented fantasy. I recommend this version (13 ep) over the longer remake personally (the other one adds more characters and goes on tangents i didn't personally like )
Steins Gate - I remember watching this years ago and enjoying it quite a bit. Don't remember what the story was about, but hey it's #3 on My Anime List, so it can't be bad, can it?

Spice and Wolf (or Merchant meets the Wise Wolf) - I watched the remake and for a romance oriented show it was very good, even though rom-adventure is really not my type, so hey that's saying something. I will be defo watching the pre-remake original seasons too.
Two modern anime titles i'd recommend are:
Frieren: Beyond Journeys End - if you do not mind the modern aesthetic and slow story it's pretty damn good. And when the big fight scenes happen they are cooked
Delicious in Dungeon - for a show that's about living off the country in a dungeon (AKA eating the monsters) it's a very good anime. Plenty of combat too.
All that i remember watching