I run an engineer on both KDF and Fed sides. I've played some form of pet handler in most every game that had such. There are thousands of STO carrier drivers with more hours than I have, and I don't even remotely claim to be an expert here.
STO carrier builds are quite flexible depending on your role, your playstyle. Your fighter layout will vary depending on the mission, and sometimes even the given target. I mostly fly tank or as a blockade ship in STFs (stopping probes, nanite spheres, etc). This is pretty much my Vo'Quv build (with some minor tweeks for the Atrox):
Weapons loadout --
Weapons loadout is a matter of debate. Being a sci ship, you want to keep your bow on the target as much as possilbe (and she'll turn like a brick). You also have target subsystem attacks, so you'll want beams to take advantage of those attacks. You are limited on tactical, so torpedo special attacks are'nt really viable. If you want to run a torp, mount something that doesnt require a special (like a Hargh'peng or Breen cluster). Given that, I run the following:
Fore: beam x2, dual beam array x1
Aft: turret x3
*If I'm running blockade, I'll usually mount a tractor mine launcher aft
Bridge Officers --
Tac(Lt): BAO1, CSV1 (I prefer scatter for peeling little targets that come your way like plasma torpedos, etc)
Eng(LtC):ET1, BP1, EWP1
Sci(Cmd):ST1, TB1, GW1,TR3
Tac(2):Antiproton Mag Regulator x2
Eng(3): Point Defense System (uni), Theta Radiation Vent (uni), Field Emitter
Sci(4): Borg Assimilated Module, Graviton Generator, Flow Capacitor (for all those yummy subsystem attacks), Particle Generator
Assimilated Subtranswarp Engines
Assimilated Deflector Array
MACO (sure wish I had my KHG shield here, nothing like it on the FED side)
Fighter Sortie --
Every carrier has a set of unique fighters that cannot be used by any other ship (Vo'Quv = BOP, Ki'Far = Frigate, Atrox = Stalkers)
Keep a pair of all the OTHER hangar pets in your inventory.
Heavies: for larger targets like gates and anything bigger than a cruiser
(BoP, Frigates are what I have on the KDF carriers, but the jury still out on how the Stalkers will be ranked but the assumption is that they will be heavy)
Power syphons: shut down cubes, baby!
Shield drones: I'll often keep on hangar load on myself and shift it to a teamate that needs them. Also great for keeping the Kang alive in the Cure.
Light fighters: pretty much most any other situation
Tetryon fighteres: if you PvP with your carrier, then they are worth the investment
In the end, there's no "wrong" carrier build as long as you're having fun with it. It can cover a lot of various roles with one ship. There are some great Vor'quv builds out there. What's your Atrox build gonna look like?