My biggest problems are with the UI. Even as I'm typing them out they seem kind of trivial, but these are the kind of games I tend to play for a while so little UI problems really add up over time.
In particular, navigating just about any list is a total pain and takes way, way too long. This is especially annoying with things like starbases or research, where you're forced to constantly go back and scroll through the lists over and over again.
Starbase upgrades: There doesn't seem to be any rhyme or reason to the way these are sorted. If there can't be some sort of separate list box for the generic weapon/defense upgrades that are present for every starbase, the upgrade modules specific to each type (economy, military, influence) should at least be at the top of the list. Honestly, the previous suggestion of having a split list box seems best, but since I have absolutely no idea how the UI code works I won't presume to think that this is an easy or even doable fix, but something, even if it's just a changing in sorting, really needs to be done.
Research: This is the big one for me. It was bad enough that I almost quit playing an hour or two after I got the game. The list of available tech upgrades, especially for someone who's new, is overwhelming once you've started down more than one path. More often than not I'm trying to research down a specific path, but I'm constantly distracted by having to scroll past a load of technologies I'm not interested in.
Ideally, I'd like to be able to pick a goal technology much further down the list and let the game silently research everything for me, but anything to reduce the amount of information overload on this screen would be really welcome. It'd be nice to be able to hide technologies from a certain path, since having ten to twenty potential technologies to research in the listbox at once is overwhelming.
Planetary Improvements: By mid-game the planetary improvements list is a mess, so much so that I usually ignore upgrading my planets as much as possible. I hate having to scroll through a huge list of improvements just to find a simple lab; it'd be really nice to have some kind of tabbed list with categories like 'basic improvements' and 'trade goods.' Failing that, just having the list permanently sorted so lab/farm/factory/market upgrades are always at the top would be a god-send.
I also like the idea of being able to build out-dated improvements, but I don't even want to imagine what a headache that would be with the current sorting.
Ship Design: The components boxes are a total nightmare by about mid-game. I'd love to be able to choose how to sort these, preferably by cost, size, or 'attribute' (damage, speed, whatever). Sorting them by name makes absolutely no sense and is very counter-intuitive. If nothing else they should be sorted by either size or cost.
Besides the UI, my other big pet peeve is the ridiculous number of custom ships I end up with. The problem is that it's tough to reproduce your exact point along the tech tree in any given game, so custom ships designed at one point in one game will probably show up far too late to be useful in another game. To use an example, right now I have about five generic railgun-with-engine designs that each show up at different times because I was at a different point along the propulsion, miniaturization, or mass drivers tech trees. All of the ships are identical except for cost.
I don't really have any suggestions for how to solve this, but having so many user-made designs is incredibly annoying and you're essentially forced into it since there are plenty of points where you have the technology to build a decent ship, but an appropriate core design isn't available yet. At the very least I'd like to be able to flag designs as one-time-use-only so tons of redundant custom ships don't appear at random times in every game that I play.
Finally, and this is more of a game balance complaint that a little tweak, it's weird having the attacker always fire first. It really reduces the strategy required when designing ships since the importance of good defense is reduced a little bit. I'd really love to see speed or sensors play into initiative, even if it's optional, because as of right now having warships with good engines or sensors doesn't seem terribly important.
