Pause often to consider your options. This is the single best thing you can do to improve your game. Because pausing is so important, I've included a screenshot of a paused battle.
All incoming projectiles will impact the exact center of a room. You can use this to your advantage to determine how to respond to a situation. You may need to move crewmembers out of a room or cloak depending on this.
Drones aim at a room before firing. Sometimes, you'll need to pay attention to what room they may be targeting. One situation this would help is when using Stealth B early on. If you're fighting a ship with a drone, don't cloak immediately. Wait till the drone is about to fire. If its going to hit a room other than weapons or cloak, you should probably wait before cloaking.
Sometimes, its best to leave a breach open. Take Federation A for example. If any of the rooms with airlocks are breached, you should leave them open because no systems can be in those rooms. Your ship will vent faster because of the breaches (Just be careful with the upper left one. If you're bringing back boarders that are almost dead, and you have people standing in the teleporter room, the boarders will beam into the oxygenless room to the left.. I may or may not have learned this the hard way). Another example is Rock B. It has no airlocks, so if one of the rooms in the lower right are breached, I usually leave it open till I have a good reason to close it.
When fighting Zoltans, if you damage a room to make a Zoltan walk through the shield room, often he'll remove a power from the system causing their shields to drop a level momentarily. Take advantage of this brief opportunity to allow one extra laser through.
If you only have enough weapons to take down the shields of an airless auto scout, but you also have a Fire Beam, you can still destroy it. When the shields are down, attempt to light as many tiles of the shield room (and other rooms) as possible in a ship. Maybe 20% of the time the fire will burn long enough that it will damage the system allowing you to get another laser through.
Learn your events. Figure out what options are the most beneficial. Know when you should take risks and when you should play it safe. Learn what races, augments, weapons, and drones grant favorable blue text events.
Crewmembers get 2-3 seconds of movement/healing/repairs/combat done during a jump. This can be used to your advantage.
Individually set weapons (like ions or Chain Vulcan) to autofire by holding Ctrl when targeting.
This isn't a tip, but here's a screenshot of some silly scientists for a chuckle.
Be flexible. if you're given a good weapon, don't hesitate to change your strategy to accommodate that weapon. If you're given some Mantis and a you have the scrap, buy a teleporter. If you have a laser build and you're given a Halberd Beam, think about switching.
Don't buy weapons that you won't be able to use for a long time. Buy efficient ones that compliment your current loadout.
Augmentations can't be stored, so sell them if they're not very useful. Unless you have great augments, try to keep one slot open in case you find one for free.
You should only accept a surrender if the offer is very enticing because you'll get more scrap for destroying the ship. If you need missiles or fuel and the offering is high, go for it. You'll almost always want to accept a surrender if they offer a weapon, drone, or augmentation because at the very least, they can be sold for scrap.
Slaver ships will usually offer you a slave before you destroy them, but sometimes they won't. They also have a good chance of rewarding you with a crewmember if you kill all the crew. If you have the right loadout, you can choose your reward. Damage the ship down to a couple hull (if you can take it down to one hull safely, do it). If they don't offer a reward, you can still board the ship to have better odds at finding a slave than if you destroyed it.
Generally don't buy weapons off of untrustworthy pirates.. You'll usually regret it.
If you see an Anti Ship Battery on the map in an Abandoned Sector, it is friendly. You'll want to try to go to that beacon because the fight shouldn't be difficult, and that event can have better than average rewards.
Exits on nebula beacons never have Anti Ship Batteries, so it may be advantageous to explore beyond the exit. But be careful, they'll often have ion storms.
Some enemy missile launchers look exactly like others. Sometimes, it will be important to try to tell the difference between them. You can do this based on how fast they charge. The best example of when this could be important is telling the difference between Hull Missiles and Breach Missiles. Breach Missiles take substantially more time to fire. Watch the red indicator light on the weapon to determine how charged it is, and you'll be able to tell which missile it is.
You should generally not be quite as merciful against pirate ships. If you encounter a pirate ship engaging another ship, destroying the pirate will give you rewards, and sometimes the ship you saved will also give you a reward.
Pause all the time to consider your options.
Learn your events, their outcomes, and what crew, weapons, drones, and augments grant good blue text options.
If you have enough ion power, and the enemy ship can't damage yours, you can suffocate the enemy crew. Even if they have a medbay, you can often still remove enough power from it to kill the crew. Just keep in mind that oxygen and medbays can be level two or three requiring more ion damage to take them completely offline.
All friendly lasers, missiles, ions, and drones can connect with and potentially destroy all non-friendly lasers, missiles, ions, and drones. Ions only stun drones with a chance of destroying them, but any other connection instantly destroys the projectile. This can be used to your advantage in times of need. If you see a large missile coming in toward the front of your ship, and one of your weapons is in line to hit it and ready to fire, you can time it so that the missile will be destroyed. The timing is very precise, so this will take practice.
Evasion is calculated when a projectile passes your shields if they're up. If your shields are not up or you don't have shields, evasion is calculated when the projectile would connect with the room it is about to hit. There are a few certain scenarios where its better to depower your shields to give yourself an extra moment before the evasion is calculated. One example of this is the Flagship missile volley. Sometimes the cloak timing is very close to the missile volley. If you depower your shields, it could buy you the extra half second you need to avoid the damage.
Ion Pulsars hit your shields first if they're on. No shields can be very bad for the health of your ship. If you toggle your shields off when an Ion Pulsar is about to hit, they won't necessarily be taken down first. However, they may still be hit randomly taking them completely offline. To get around this, toggle your shields off, then put power back into the system, but don't leave a full level of shield build up before the pulsar hits. This way, even if your shields are randomly hit by the pulsar, you may still have a level or two of shields to work with.
You're not pausing enough. Pause more.