A lot of the big ones have been said but I'll try to touch on everything that's been in my head:
On combat: tactical options for fitting your ships
Right now, there's no reason to choose one type of weapon over another aside from trying to exploit your enemy's weakness. Ships fitted evenly with 1:1:1 ratio of weapons don't really do much more than ships fitted with 1:1:0 or 1:0:0 except for the fact that they each "roll" seperately and might do slightly more even damage. Ships sizes have nothing to do with how maneuverable they are or how easy to hit they are, sensors are worthless in combat and especially worthless after eyes of the universe.
Ships that are 2 sizes smaller than their opponents receive a bonus to evade, a bonus that increases exponentially with size reduction. Because of increased mass larger ships would rarely move as fast as small fighters as they cannot stop/alter their courses as easily. This should be represented visually and by the aforementioned negatives to hit smaller craft.
Sensors offset these negatives to a degree. The more sophisticated the sensors the more easily larger ships can track and strike at smaller craft with fixed guns.
Each type of wepons system has special attributes. Beam weapons are the most accurate as they travel the fastest through space but do the least amount of damage. They are best for smaller ships. Missiles track and carry a fairly heavy payload so they are the all-around weapon that works well against most ship types. the drawback could be that since they travel slower, other weapon types would always receive the first strike. Mass drivers would be the most powerful weapon type but very innacurate against smaller ships. It would be the main anti-capital ship weapon.
Each type of weapon has a HEAVY and LIGHT version for dealing with different threats. specializing in one weapon would actually punish your empire in the long run as you'd have a harder time vesus differnet types of ships. It also makes no sense that a certain type of weapon only has one configuration. Essentially, the heavy version of a weapon has its own damage bonus and accuracy penalty, whereas the lighter version does less damage but is more accurate. This doesn't negate the bonuses of the seperate weapons systems though, a light version of a mass driver is still less accurate than a heavy version of a laser.
Defenses have hitpoints rather than armor ratings. Having a group of 1 attack fighters against a huge ship with large defenses still would mean nothing though, as they're unlikely to penetrate their defenses. Having shields, armor and point defense decrease with use however woul allow smaller ships to wear down heavy defenses rather than bounce harmlessly off. Without this option, there really is no use in still having smaller ships, as they mean nothing versus larger ones.
These options already give a far greater range of tactical options when fitting ships. Will that tiny hulled fighter be equipped with lasers to deal with enemy fighters, or mass drivers to be anti-capital ship support? Will you make enough room in a capital ships hull for lasers and sensors, or will you deck it out with huge mass drivers to annihilate enemy battleships? Will some ships be created to specifically hunt other ships or will many be readied for many different threats?
Tactical combat = probably won't happen, so allow the player to set "priorities"
Rather than creating a real-time combat system, players would be better served with a scaled back way to interact with battles that doesn't put them in the role of admiral or captain, but still allows them some control. Before battle commences a screen would appear that lists the ships in the battle and allows the same type of "intelligence report" that you could see in any other part of the game. The player can then select individual ships and set their "priority" from a drop down menu. These priorities could be:
Huge/Large/Medium/Small/Tiny ships
Strongest/Weakest ships
Ship with armor/shields/point defense
Ships with modules (troop ships, carriers)
This way you can seperate your anti-fighter fighters versus your "bombers," Your anti-fighter gunships from your heavy attack gunships etc.
Cloaking, detection and spy ships
in an effort to make sensors more useful after eyes of the universe are created, a ship stealth tech tree could be available allowing the use of a cloaking module. This would allow for true stealth ships and stealth attack ships. the way it would work is that there are 5 levels of cloaking and 5 levels of sensors. If the cloaking level of one ship is higher than the detection level of the other ship (adding more sensros doesn't help) then the ship is not detected whatsoever. If the sensors one one ship is more advanced than the other ships cloak, then it is detected. If they are even, then either the ship pops in and out of view, or the enemy cannot determine the size of the fleet or strength of the ship. This is important when dealing with my last point:
The AI assigning strength levels to certain sectors, fleets and planets
The AI as many people say isn't the best. What I think is the worst part is that the AI doesn't properly attack or defend their land with their ships. As soon as war is declared, enemy ships head straight for their targets single mindedly, no matter where they are. The Ai instead should assign importance to certain planets, and assign danger to enemy fleets:
Planets with high production, research, economy or with special trade goods or improvements should be seen as more important and more worthy of defense than small class 4 planets with no real purpose. Enemy fleets that are large and well armed, or troop transports heading to undefended planets should be seen as more of a threat than weak fleets or single ships.
With this setup, the AI will be quick to rush to the aid of an important planet that has even an insignificant enemy force near it and would respond by sending a large number of its forces to intercept. A constructor or freighter would receive practically no attention while this is taking place, unless there are a few ships close to it that aren't needed in the battle. this means that the Ai properly defends it most important outposts and planets while trying to destroy the players most important fleets. this can also mean more strategic depth for the player: sending a moderately sized fleet to an important planet would draw enemy forces to it, while another moderatlely sized fleet with a troop transport can attack and take over a smaller world.
Anyhow, those are my thoughts for now. Sorry if there are spelling mistakes, for some reason spellcheck isn't working with firefox anymore...