Wow... This hurts me. I understand tactical should have better combat skills than others, but beyond that, department has nothing to do with combat. Turning phaser battles into traditional MMO fare hurts my heart. I would hate ground combat if Command was buffing, Science officers were debuffing and Tactical were nuking. It's broken as is, and its certainly broken for Trek. That kind of combat design paradigm doesn't work without damage soakers, and seeing someone out there soaking up phaser fire with their "forcefield" would really kill any Trekkiness I had accrued on my PSI. This way madness lies: "Can't have Ship's counselors... unless they can mindmeld/betazoid/ubertalk the enemy into a stupor."
As for what to do planetside, planets will need to be made up of multiple kinds of challenges, rather than just be fields of mobs. Again, imho, Star Trek Online cannot be Trek without challenging, engaging, rewarding non-combat gameplay.
As for Bridge Missions... there are multiple levels to it. All subsequent text is from my dream world:
The First: Ship to Ship Combat
The first deceptive thing is that everyone involved in ship to ship combat isn't necessarily on the bridge. Engineers, the 'healers' of this group are constantly running around ship. On larger ships it could be dozens of PCs. Science Officers, perhaps analogus to the "nukers" are crafting up some sciencey solution to cripple or end the threat. If the solution is multi-part and requires basic solutions to create an advanced solution, this could be a half-dozen PCs. If Intruders and Invasions are involved routinely, as they should be, then the ship is either full of combat while this is going on, or players are porting over to the other ship during breaks in the shielding for combat. This could be dozens of PCs as well.
And then you have the actual bridge. Obviously the Helm Officer is in full sim mode. The Tactical Officer is queuing attacks and managing the nuances of the shields... what type of defenses basically are being put up, basically. The Captain, and 1st and 2nd officers, and Operations Officer are the buffers here. The Cap, 1st and 2nd are sending NPCs to help either Tactical, Engineering, Medical or Science, making decisions similar to the ones an RTS player might make. The Commanding officer is also issuing 'commands' to specific players. That is, buffed versions of their own moves, or perhaps moves that they don't have access to/haven't unlocked yet. Players have the option of bypassing these commands, however, its rarely wise to pass on free buffs and more XP for all. Operations Officers really just move power from one 'system' to another, for various effect on the ship. Possibly Ops Officers can control forcefield and doors in the ship in order to control the intruder combat via a bird's eye camera, but lets not get greedy. But lets face it, the actual 'combat' is being done by the Tactical officer and helm officer and being overseen by the Captain. If you want to do Trek, all that supportive stuff has to be gameplay.
The Second: Exploration/Anomaly Missions
Well, sometimes you're on the bridge, but you're not fighting. Novel Concept I know. So what the heck are you doing? Honestly... you're doing all the other stuff on the ship and taking care of downtime activities like training up skills, socializing, repairing long term damage to the ship, feeding/training your NPCs, etc. Let NPCs man the tactical station while you're away. But what if you just HAVE to be on the bridge? Well... you're grinding.
Of course if you're the CO, then you've been grinding for a while, explore this way, explore that way, this heading that heading, which mission should we pick up? Etc... For the truly bored a Comm Officer or Sensors Officer could scour feeds for useful information, perhaps even find missions this way. Again, limited grinding with adequate reward.
But then an anomaly pops up. Some strange unexplainable thing that's having an effect on a planet, another ship... or yours. Now we have a challenge. Science Officers have to figure out what to build to get around this. Captains have decisions to make. Engineers have to make compensations. Tactical may even have to figure out new shield configurations. Meanwhile the missions relating to the entity have all been generated. You never know when an anomaly is going to drive some unique crew member crazy, y'know?
That's how I picture bridge missions, both involving the whole ship and being varied enough that "Bridge Missions" isn't really a useful term.