I hate to say it, as it feels like giving away trade secrets from a drone builder but, I find the following methods to be useful in countering drones...
1) Loads of AMS. Most drones attempt to maximize firepower by using missile based weapons. Only a handful I've seen successfully use beams/cannons to any extent. Hence AMS will neuter most drones, and why most of my drones are now cannon/beam based. Make sure it is plentiful, around 10 AMS turrets should suffice (at the bare minimum), otherwise they will overload your AMS and you won't get any use of it against the primary enemy vessel.
2) Pulse drones are annoying but as a player ship you can safely ignore them in favor of virtually anything larger. AI will need buried turrets to fight against pulse weapons. By placing the turrets behind layers of the ship's armor(not the turrets armor) less of the turret will be exposed to the fairly random pulses that are fired off. It'll take more internal space but I feel it's worth it given the benefits. I'll share an example pic later...
3) Push and Pull effects get very sloppy very quickly, if you're going to use an effect based turret, use either EMP or Stop effect. Drones can't carry a ton of energy given their size and most are barely considered power stable, any disruption to that is catastrophic. Stop effect also has a greater effect given the low mass of the drones so even smaller turrets can effectively neutralize pulse drones. Alternatively overdrive turns a rather small weapon into something considerably more potent allowing smaller turrets to perform better against drones with armor.
4) Never use swarm missiles, they are barely more effective than regular missiles (and arguably so). One would think they have a greater effect but they target anything that includes your own turrets, your own ship, any astronauts, random space debris, chaff drones without any AI at all, and the list goes on and on. Odds of them actually hitting the intended targets is pretty slim. With newer models having AMS that makes this option even less effective.