We are not responsible if these tips cause your ship to implode/explode. Contains spoilers in vast quantities.
Use Cloaking after enemy weapons have fired so that they miss.
Upgrade Shields as soon as possible; many early-sector enemies can't get through 2 shields.
Any of your crew inside the medbay will be healed, they don't have to be standing still. With a lot of pausing and fiddling with crew movement you can heal 4 crew in a 3-slot medbay this way.
Upgrade your Doors early on if your ship has trouble dealing with boarders. Higher-level doors slow down boarders trying to break through them.
2 Zoltans in a shield room leaves one unionizable shield layer - Zoltan-provided power ignores ion damage.
Upgraded sensors can show you where crew are and even weapon charge levels, and thus may be worth considering.
The enemy AI appears to consider Shields the most important system and will prioritize this over other many things.
With the Rock C, naming the Crystal crewmember you start with "Ruwen" gives you a quest marker in the Rock Homeworlds. This leads to the Crystal sector portal.
Upgrade your engines!
You can teleport mind controlled enemy crew onto your own ship, and then kill them.
Dropping your shields before an ion blast hits might make it hit somewhere unimportant instead of removing a shield layer for >=5 seconds. They do take some time to come up again, though.
Killing the crew of a ship - usually by boarding, but there are many other ways - gets you more loot than destroying it.
A Defense Drone I can often be better than Cloaking - assuming you have Shields.
Firing two missiles can overwhelm a single defense drone. They also try and shoot asteroids.
To unlock the Crystal Cruiser:
Get the Damaged Statis Pod from a Dense Asteroid Field.
Find some Zoltan scientists asking to take readings, and get them to open it.
Go to the Rock Homeworlds.
Find the quest marker and activate the portal.
Find the quest marker in the Crystal sector. Then you get the ship, along with some loot.
Brag to all your friends, shouting things like "HAHA, I unlocked the Crystal Cruiser!"
Most stuff doesn't need to be on all the time - oxygen can stay off for a while without horrible problems, you barely ever need your medbay and engines/shields only need to be on just before a volley (though shields significantly before). Basically, be good at power management and pause lots to optimize this.
Upgrade your engines more! Also subsystems and stuff like oxygen; the upgrades can often be unexpectedly helpful.
Don't constantly stay 100% repaired (from shops) as free/cheap repairs come along frequently.
Hostile sectors provide more loot too, as there is more combat. This also, obviously, makes them more dangerous.
If a ship jumps away, you can jump before it does and skip the penalties (they often lead the fleet after you, etc).
Beam weapons cannot miss, and can also go through shields if they have one or more damage per room than the ship being targeted has shield layers.
Pausing to think or plan, or really just all the time, is a good idea.
If a ship is incapable of actually damaging you, you can use it to train your shield/engine/pilot operators (sometimes weapons) without manual interaction to make them more effective.
It's a good idea to save some scrap for emergencies or random good things you may encounter instead of spending it all at stores.
Visit as many beacons in a sector as you can without the fleet catching you.
Fire weapons in volleys so that stronger ones can hit the enemy ship after weaker ones take down shields. Autofire is bad for anything but ion weapons.
Buying crew at stores is generally not worth it, unless you are really low somehow.
Do not waste missiles. But also do use them, as missiles are often cheaper than the damage you might sustain if you avoid them too much.
If your crew are not good fighters, you can open the doors to rooms containing boarders to vent out the oxygen, so they will slowly die. They can break the doors, though, or go through them freely if your doors are only level 1. If the situation is very dire and you aren't in ship-to-ship combat it may be worth moving all your crew to the medbay (or rooms adjacent to it, if there's not space in there) and venting all the ship except that. Of course, if the boarders are damaging your oxygen or door system (or just close to it), this is a bad idea and you should not do it.
Don't jump to stores if you don't have things to sell or lots of scrap. You're missing out on scrap you might otherwise get (this is called "opportunity cost").
Don't expect to win all the time or even particularly often. FTL is a roguelike. It's designed to be played repeatedly, not won all of the time, and involves heavy randomness.
Ships' starting weapons are actually generally pretty good, often using less power and/or firing faster than purchaseable alternatives.