How can I defend effectively against missile weapons?

There are three ways to avoid missiles, each with differing costs:

1. Upgrade your engines (hard to do early game, but at least keep both the engines and piloting manned to boost chances of avoiding them).
2. Cloak, if you cloak as soon as you see the shot you can avoid the missile. You'll need the ability to cloak, which you have to purchase from a shop.
3. Defense Drones (and also Mark 2 ones) will destroy incoming missiles but you'll need a drone bay for your ship if you don't start with one and also find the right drone schematic.

1 is the easiest. 2 & 3 requires finding at least one shop, and 3 requires additional luck if you don't get the right drone schematic when you purchase the drone bay. Early game, unless your ship starts out with a drone bay, you may just have to take your lumps.

Early in the game you should be focusing all your attacks on the enemy's weapon system, to shut it down as quickly as possible. If they can't shoot missiles, you don't have to defend against missiles.

Kestrel: For the entire first sector, and usually the second also, always target your Burst Laser at their weapons systems until they are red, and usually keep targeting after that unless they have drones. If the enemy has multiple missile launchers, fire your Artemis missiles at their weapons as well as your Burst Laser. But if they only have one, don't - keep your energy in the engine and conserve your missiles.

Torus: Alternate your ion shot between shields and weapons. Get a Defense I (or II) drone as soon as possible. This ship is just plain vulnerable at the start.

All Ships: Upgrade your engines quickly, to 3-4 bars at least. Prefer to keep an initial crew member in the engine room, not the shield or weapons rooms. (Usually I leave shields empty if I start with < 4 people in my crew.)

About drones: Drones are great for defense, eventually. But in the early game, they're more expensive than they are worth. A drone control system, a defense drone, and the energy to power it will cost you 170 scrap. If you're trying to reduce hull damage (which is a good first-approximation strategy), that kind of scrap is much better invested in shields (level 2 if you start with shields, the system per se if you're using the stealth cruiser) or engines (you can get up to level 5 and change).

Plus it's fundamentally unreliable - if you don't find a store with the right kit, this whole strategy falls apart. You'll have found that by sector 5-6 sure, but by then the cost of repairing hull damage is also comparatively less annoying - you need to make sure you don't start falling behind right away at sector 1, and drones just aren't feasible for that unless you're in a ship that starts with them.

In rough order of "how fast you can implement this".

1. Don't get hit. Upgrading Engines (and staffing them along with Helm) increases your dodge change. Cloaking is great if you have it (pause when you see the missile launch for best use).
2. Remove their ability to fight - target and destroy/ion their weapons. If they can't power the missiles, you can't be hit by them.
3. Active defense - Drone system, a Defense I or II drone, and two/four points of power. (Plus a supply of drones). It's probably the most reliable once you have it running, but it takes more setup and ongoing resources. (Also worth noting that you can overwhelm missile defense by shooting enough missiles at once).