Of the things I want most, I would probably say having them fix the Galaxy model is the biggest thing bothering me atm.
Other than that, I would say adding in solar systems instead of just a random planet layout when you enter a system. I had always thought that this was what the genesis system was supposed to do, not just create a single planet and a moon/debris field so close in orbit that it would crash into the planet within a few days rather than orbit it at all.
On that note, the stars in those systems should look more like stars instead of a glowing bubble. I've flown into systems where the planet looked more like a star than the star did because of how much it was lit up.
I would like to see the ship's interior added. Make your ship feel alive! With off course pirates who try to steal your cargo, just several men, not 6 six rooms full (you need to complete the mission within 5 minutes otherwise you will fail -> leads to some money (energy) loss...fun! )
Outside of Nimoy, of course. Quinto sounded like he only had half an hour available, so they just threw him the script and he got out of there as quick as possible without actually pre-reading it. I only say this because I thought he had a great performance as Spock. Compared to the EMH, not so much.
I, too, would like to see more of my own ship. And missions that take place aboard it. "Captain, Klingon boarders have taken over the Engineering section. We need to go down there and get them out!" Make it part of a sequence of missions rather than something that comes up independently.
More Star Trek music. (I bet licensing fees would be expensive, though.)
I like the fleet headquarters idea. Make it customizable, like bases in City of Heroes. Give it some uses, like exchange and bank links, uniform customization, etc. (Of course, it'd be a while before we saw something that complex.)
Explore missions that aren't just "Click on A. Now click on B. Now click on C." Throw in a few obstacles to overcome.
"Stun" settings for all weapons. There are times I'm not comfortable vaporizing my opponents. (I know, just don't eve use Exploit Attacks. But man, some of them are also simply good attacks!)
Different colors for Federation and enemy weapons. Not full customization, just something to add variety to the appearances. It wouldn't take long to learn what represents what, and it would fit the canon better. (OTOH, it is nice to know Green = Disruptors.)
More high level fleet actions that work, the ability to start a fleet action as a fleet instead of having to jump groups of 5 around until you get everyone finally in the same instance. Romulans after Klingons are in a much better spot obviously.
I know the OP said "besides fixes" but grouped missions are so horribly buggy that I don't even like bothering
Second: More diplomatic and exploration missions. And more than just "Click 5 glowy things" I want something like what happened in the TV shows. I got a mission that was about fixing some radiation poisoning in a pre-warp civ. I didn't get the choice NOT to fix it (Prime Directive anybody?) and it really was just "click 5 glowies".
I'm still enjoying the game though but these are the two things I want the most. Also, get Shatner to come in and do a voiceover for the Guardian mission because I think Kirk would talk to you too, not just Spock
A large and open PvP zone that will allow different factions to gain control of variuos systems, sectors and the like so we can have a player controlled war setting. Not to mention proffessions that would allow a player to function in such an enviroment. ei: smugglers, spies and the like.