On Navy ships do the first officer's really spend much time on the bridge? I've heard that they basically run the ship for the captain and that they're all over the place so he doesn't have to be. Is that accurate? Because if it is it's unlikely Riker would have been on the bridge at the same time Picard was.
It depends, their is an XO and First Officer, sometimes they are the same person. Sometimes not.
The XO (Executive Officer) set's duty rosters, discipline aboard ship, chow rotations, PT times and types etc etc etc.
The Captain sets the standard and expectations and has complete charge over everything.
The First Officer acts in the Captain's name and does the same things the Captain does usually during a different duty shift. Can be and "often" is the XO too, but does not have to be.
The Second Officer is the same as first officer, only to a lesser degree. They are usually not the XO as well and have the least desired duty shift but do have command of the ship.
When second and third officers and sometimes even first officers they have a limited amount of discretion. Whereas the Captain has the final word and must stirred and briefed and he makes the call. (ie if a USN ship has the 1st Officer in command he cannot make the call to fire on a ship that may be dangerous but not currently firing on the USN ship, whereas the Captain can just blow the thing out of the water)
I remember in some TNG episodes Ricker is discussing things of the XO field of work so he is also the XO and First Officer. In 'reality' you would not often see the Captain and First Officer during the same duty shift.
This is not a set in stone Navy SOP but what I have gotten from some of my Navy buddies after all I am just a groundpounder.