As far as I can tell, some things require navigating a labyrinth of screens to do (e.g. trying to figure out how to design fighters -- I tried, but they aren't available for construction as anything but ships) and/or are very tedious (e.g. building a large ground army and organizing it into divisions/brigades). I'm having a lot of trouble trying to figure out task groups, and I'm not even sure I
can get them to do what I want without lots of babysitting (e.g. have asteroid miners go around mining asteroids without my input -- I think civilian ships can do that for me, so I've tried giving large subsidies to my shipping line).
There are also a lot of things that could afford to be automated or at least partially automated. I'm horrible at ship design. I keep making oversized, slow-moving monstrosities (may be a result of researching oversized engines and such?) compared to some of the player designs I've seen floating around. When you start a new game and/or empire, you can say "use some of my research points to create ship designs" and the game will design some ships/parts for you. It'd be nice if we could continue to use that ability by clicking a "randomize this design" button (bonus points if it fits a hull type of our choosing, like Freighter or whatever). Then we can find something that looks good and tweak it as needed without having to design it from scratch (i.e. it'd be an effective guide for those of us who are clueless). The same random design feature could apply to ship parts, missiles, and so forth.
As for the army organization I mentioned, it'd suffice to be able to select multiple battalions and click a button that organizes them into brigades automagically as best it can. Want a garrison brigade? Just select some garrison, brigade, and division units then click the button.
Stuff like the latter that could really help take some of the tedium out of the game. Stuff like the former would be not only convenient but would make the game a bit more newbie-friendly. A more thorough wiki would also help with that last point.
Another thought: Flag certain features of an empire for pure AI control, as though it were an NPR. Don't want to bother with research right now? Let the AI handle it. Similarly, the AI could handle military organization and/or maneuvering with just a few macro-level controls for things like "do/don't explore new systems" and "feel free to attack this or that empire". Or it could handle colonization or infrastructure/installations within any particular system. Stuff like that.