You can't "pilot" the carrier, if it is caught alone (without fighters) two things can happen: If the enemy fighters have no torps, they get bounced back to the system they came from, if they DO have torps, the carrier is automatically dead, sector is over.
If you have fighters docked or escorting the CV, you have to choose one.
There is a bit of story, the Voices of War manual is one of the best, and it tells the story of the two self-sustained Supercarriers, the confed one and the kat one. Once the game starts, though, it is just plotting and fighting.
There are transports, they are used to ferry the abstract "resources" from planets to the carrier - which is the only ship capable of building mines, shipyards and fortresses. Lone transports were automatically captured by enemy fighters... But these 'sports were usually safe behind your own lines... And no, you cannot fly the 'sports. You had to keep your CV supplied with "resources" so you could build up your empire. The more mines and shipyards you had, the more fighters you got. And fortresses were useful to close down certain chokepoint systems you found on every sector. Fortresses were actually bases with 3 regenerating fighters. There were 3 types, light, medium and heavy, light being stocked with light fighters and so on. Of course, if you lost ALL 3 fighters, you lost the fortress, but if at least one survived, the next round you had 3 of them again. The advantage of this is that you didn't need to build these fighters and could move on with your CV while keeping your mines and yards protected.
The main difference between kats and confees were the fighters. You can see them for yourself in the CIC's shipguide, though.