There is a lot of advice scattered around various topics, so some of this will inevitably sound familiar, but I suppose it's safe to share some things I've learned from experience. Some of this information certainly won't be found in the in-game list of tips. Please excuse me if it gets long-winded! If I think of anything else, I will add it later. Feel free to add your own tips, which may be incorporated into the original post for convenience (with your permission). In general, the following advice is universal, not ship-specific, and does not include any spoilers regarding random event choices or the final boss. :)

1. Explore.
Don't rush through each sector. The more beacons you visit, the more battles you fight, the more scrap you will earn, and friendly encounters may even give you a weapon or augment or new crew member for free. Ideally, you will want to exit each sector just barely before the rebel wave catches you. Remember, though, if you DO get caught, you must fight an elite rebel fighter, which earns no rewards and can sometimes be devastating. Don't risk your ship over greed to explore one more beacon.
-- Don't worry about fuel too much. Battles and random events often drop fuel, and you can buy more at a store if you're low. Keeping a dozen fuel units in storage whenever possible is a good idea, though, as it can preserve you from a temporary dearth due to bad luck.

2. Things to have early on:
-- One extra shield bubble early (requires two shield system upgrades) can save you a lot of repair bills later. Just remember it will need reactor power too.
-- One or two additional crew, preferably all of different races, can greatly boost your resistance to boarding, and your system repair and fire-fighting ability, as well as allow you to use racial traits in special events.
-- Upgrade your ship's doors at least once. Blast doors hinder fires and intruder movement, allowing you to trap them, and giving you more time to react effectively. Both can even be suffocated by evacuating the room's oxygen through a nearby airlock. Without door upgrades, the greatest danger is small fires getting out of control while your crew is distracted with intruders or repairs elsewhere. Always respond to fires promptly, either by venting oxygen or sending multiple crew to extinguish them.
-- Start training each crew member to a specific task or ship system early. Keep all systems manned whenever possible, and always with the same person. Of course, it's inevitable that they will sometimes have to leave to help with repairs or combat. That's okay. And if they must go to the medbay, or if they die, having a rookie on the controls is better than no one at all. Just remember the veteran won't take the controls upon return unless the rookie leaves the room. It's also better, if you have extra crew, to have idle rookies moving to hotspots around the ship and repairing things, than to interrupt a veteran's work.
-- Humans have no special abilities, so if you have any human crew, make one of them your pilot - they'll rarely leave the cockpit, because the ship can't fly itself or dodge by itself without autopilot upgrades.
-- Having just one engine upgrade increases your dodge chance, which will also increase your pilot's skill faster!

3. Keep some scrap in savings.
Beyond the aforementioned things, don't spend all of it on ship upgrades immediately. You will find stores throughout your travels that may be carrying something you want to buy, like extra fuel and shiny new weapons. Everything costs scrap. Cloaking, one of the most helpful systems (though not absolutely necessary), costs a whopping 150 scrap, so start saving up! You can also sell off unused equipment or equipment that you'd like to replace with a new purchase.

4. Pause often.
This will give you time to think and plan, and to react each time the enemy makes a move. You can change your weapons' targets, reroute power, open/close doors, and move crew at leisure.

5. Power can be rerouted.
Not all systems need to be fully powered at all times. The medbay and engines can especially be switched on and off when needed.
-- Try only powering the engines when actively dodging incoming projectiles - or unless you are clearly outgunned and need to quickly charge the FTL drive and escape.
-- You can cut the oxygen for a few seconds to give the engine a slight boost when dodging as well. This won't hurt your crew. Just be sure to turn it back on! If the oxygen system becomes damaged, also check on it after it is repaired. It won't always power-up automatically.
-- You can even de-power one of your shield bubbles if necessary. Turn it back on as soon as the enemy fires a laser, and the shield will have recharged by the time the projectiles hit.
-- Cloaking can be kept unpowered until the instant you need it, however, power cannot be removed from this system while it is recharging.
-- A particularly fast-charging weapon can be worth deactivating for a second or two as well.
-- All that said, don't spend an exorbitant amount on reactor upgrades, but nonetheless be sure you have enough output to power oxygen and one engine bar (plus at least one shield and a weapon, if you wish to fight), should you get caught in an ion storm. It will happen eventually.

6. DO NOT always use autofire.
Timing is important. If you have more than one laser or ion weapon, allow them all to charge up, then pause and manually target the enemy ship with all of them. Firing everything at once like this will overwhelm the enemy's shields instead of allowing them time to recharge between your volleys. If you have a beam weapon, wait for it to charge, then create an opening with your lasers or take out the shields with a missile or bomb. Pause before any of their shield bubbles recharge, and draw the best line you can across the enemy's hull.

7. Some weapons are more power-efficient than others.
A Burst Laser Mk. III may fire five bolts, but it needs a lot of power, and by the time the fifth bolt hits, one enemy shield bubble will have recharged. Two cheap dual lasers can do the same job for less power, BUT will occupy more equipment slots. This has not been a problem for me.
-- The Ion Blast II (comes standard on Torus) deserves a special note. It is one of the few weapons that benefit from autofiring. The 3 power bars needed to operate it are well worth it. If manned, it fires fast enough to wear down even the most powerful shields all by itself. It takes time, but if you too have good enough defenses, this can be a workable strategy. Keep the ion targeting the shields to keep them down while a drone or a second weapon tears into the hull. Once the shields are fully disabled, there's actually time to fire one or two ion bolts at different systems before resuming bombardment of the shields.
-- Missiles and bombs, since they ignore all shields, are also useful for knocking out shields and delaying enemy weapons. Just remember they use ammo, so use them sparingly and don't run out! Enemies use them too. They are the reason you will need to upgrade your engine for dodging, not just your shields for blocking. An ion bomb only uses 1 power and does no damage, but will disable fully-upgraded shields (or any other system) for several seconds, allowing a devastating barrage from your other weapons. Other types of bombs can injure crew or start fires.

8. Sometimes accepting surrender is profitable.
You'll usually get more scrap for simply destroying the enemy ship, but when they get scared, the items they offer might be more valuable than the scrap, depending on your situation. Surrenders often offer quite a lot of fuel, missiles, or even a new weapon or crew member.

That's all for now. I hope it helps!

Good tips. I've figured most of these out from playing but I like some of the power managing tips.

Very good advice! Even Dr.Zoidberg should be able to win the game using these.

Excellent post.

I think number 4 should be number 1. Pausing is extremely important in this game.

Some advanced techniques absolutely require pausing.

1. Crew shuffling
Example: Crew1 and Crew2 needs to be swapped because Crew1 is dying (fighting enemy crew for example), while paused, just tell both to leave the room, tell them to re-enter the room in the order you want them positioned, unpause and they'll switch spots. You can absolutely do this without pausing but your crew will be running out and in the room again, which is very inefficient and could be detrimental to your success.
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2. Healing more crew than positions in the med bay
Running crew through the med bay heals them. You can have 3 standing in the med bay and run the other 5 back and forth through it to heal all 8 at once. Useful when O2 is low or when fire is all over your ship. I saw a video of someone who had an event where he had no O2 for a battle and managed to keep every single crew alive by using this technique.

3. Turning off shields and turning them back on to avoid ion damage
Very situational but can be useful early on, for example when you have 1 shield and they have 1 laser and 1 ion weapon. Requires very high precision and is very difficult without pausing. Also knowledge of enemy weapons is useful to recognize charge times. From my experience, mostly can't use this technique on the enemy's first volley, but after their weapons start shooting out of sync, this technique really helps.

4. Advanced techniques in allocation of power
Example1: Since shield only recharges one at a time, you don't need to power all 4 shields if the enemy shot down all 4 shields; you just need 1 extra shield powered at a time for it to recharge. Specific example: you have just enough reactor power to power 4 shields but only 1 bar on engines. Enemy shoots barrage of 7 lasers. As each shield goes down, you can convert that power to engines to have a better percent chance of dodging the next attack. After the barrage, you can recharge your shields.
Example2: Additional power on engines just for enemy missiles. In a battle, if your shield is enough to withstand enemy lasers, you can allocate power from O2, tele, med bay, etc. just at the moment that the missile would go through your shields. You just need the engine to dodge at that moment, then you can reroute power back to your other subsystems. If your shield isn't enough to withstand lasers, you can use example1 and also combine it with this example.

5. Teleporting onto unmanned ships
You can safely teleport and return from unmanned ships with crystal and rock crew with level 1 teleporters, anybody with 100HP with level 2 teleporters and even Zoltan with level 3 teleporter. This is very dangerous so you must pause often. With level 2 teleporter and starting at 100HP, your crew will be down to 9-10HP once your teleporter cools down. Also you must repair right away if they shoot down your teleporter. Be careful of enemy cloak and also if the enemy is charging FTL.

6. Distracting enemy crew by running crew around on their ship
If they have level 1 doors or disabled doors, you can distract 4 enemy crew with 2 crew by running them around the enemy ship. You can distract up to 6 enemy with 3 crew and so on. This requires high level of micromanagement as you will most likely be distracting them while preparing something else. Pausing is a must. Useful to get the enemy pilot away to improve your chance of hitting with weapons or delaying enemy FTL charge.

That's all I can think of for now. I would love it if this becomes a discussion of our techniques, builds, etc. Love this game to death. My Raptr profile shows that I've been playing this game for over 237 hours. Still enjoying it even now. Had an issue once where FTL crashed and lost my saved profile but didn't matter because I love playing it anyway. Of course, now I sync up my save to dropbox so it won't happen again!

Edit: added formatting and changed some wording for readability.

Power management is most common (for me) when using drones, especially defence drones Mk1 (I actually hate the MkII since they'll often shoot lasers in favor of missiles!). I typically launch one, but only keep the power active when there's a missile coming my way (taking power out of engines or maybe shields if necessary). Once the missile has been dealt with, switch power back to shields and/or engines to block/dodge the other shots.

Also, when something gets damaged is a good time to switch power around while the damaged power-sucking unit gets fixed - no point in wasting those power bars that used to power your weapons when they could be going to your engines. It's obvious I know, but it's easy to miss in the heat of battle.

Finally, if you practice good power management techniques then you can usually put off buying more reactor power and instead buy more shields/weapons/stuff in stores/etc.

Oh yeah forgot about the defense drone. I think the general consensus is that the Mk1 is better because Mk2 takes twice as much power and at times will be distracted by lasers.
I usually just keep the defense drone on now because I always forget to turn it on when missiles are coming. I can hear lasers better than missiles so I usually adjust power for lasers than to adjust power for missiles. Same concept as you but reversed. Either way, you are right about putting off buying reactor power. It allows you to diversify early to more different situations when you upgrade your systems or buy weapons/augments/systems. One thing I would like to say for new players is: don't be afraid to turn off O2 if you know you can win the battle before your crew asphyxiates, or you can toggle it on and off depending on situation.

This is especially relevant in those annoying nebula ion storms, although in these cases the decision to turn off the oxygen is often made for you.

In these nasty ion storm situations you may have to make some tough decisions and/or lots of power micromanagement. Exactly what you prioritise and when will depend on your ship and the enemy's ship, and what they have activated. Remember that the enemy will be affected by the ion storm too, and will sometimes shift power priorities as the situation changes (e.g. they may activate more shields as their weapon systems take damage).

Just remember to turn it the oxygen back on again! Though I guess it's difficult to not notice the beeping/flashing warnings that come with running low on oxygen.

My skill at FTL recently went from incompetant to below average, so the learning curve is still fresh in my mind.

I would urge bitzerker to keep "explore" as #1, because it's the hardest to figure out for a new player. It was the most valuable advice for me (from someone I know who played) to go from struggling to able to play properly. Without knowing to explore, a player could skip encounters (as negative events) and just dislike the lack of credits. With exploring, it becomes clear there's a lot more time fighting easier enemies and gaining credits.

I would add to the "Explore" advice that the rebel fleet is just one ship with a meager 1 fuel reward. It's not cost effective, but it won't kill you. The rebel fleet turns all encounters into bad deals, but survivable ones. So reach as many locations as possible early on, because each sector upgrades the enemies you face.

The map shows how far the fleet will move by the next turn, so if you assume it will advance at the same rate every turn (usually the case unless an event increases or decreases this) you can usually figure out where you can get to before they catch up to you.

I agree that it's generally worth the risk to have the fleet catch up rather than miss out on an extra jump point - plus killing them still counts as a kill, so it ups your score.

It also helps if you turn on the option to display where jump points lead to (shown when you hover over them).

There sometimes options in events that allow you to delay the fleet, but personally I find them not to be worth the price in scrap/supplies.