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Aside from the usual classics, does anyone have a recommendation for a more serious, less cartoonish series? Mecha, Sci-fi, or military elements are a plus.
First thing that comes to mind is Code Geass, it has every element you mentioned. It has mech battles and is about a rebellion in a empire. So you see both the different aspects of being both in the rebellion and in the imperial army so that covers the military aspect. There is also a fantasy aspect that they take a scientific/strategic approach to. There are a few lighthearted moments but the serious stuff is what you'll remember, I promise. It also has strong cat and mouse elements, because the empire is trying to locate/identify the protagonist.
Aldnoah Zero is another show that comes to mind, it's got mechs, humans from mars(Sci-fi), and is about a war between the earth humans and the martian humans. It's a shorter show so it stays serious the whole way through. I haven't seen the 2nd season but I did like the first season.
Also I haven't seen it, but Knights of Sidonia I think hits all your elements. From what I understand it's about mechs in space and is about fighting aliens.
If you want a full adult themed serious anime checkout either Elfen Lied or Berserk, Elfen Lied is about a human weapon that breaks out from the military and gets super crazy in terms of violence and gore. Berserk is about medieval mercenaries and has full portrayal of medieval warring. Both of these animes are very adult and very serious.