It seems like every time I die I find it could have been avoided had I upgraded a certain system. Are there some systems you should upgrade early in the game? Are there systems you should ignore until a certain point later in the game? They all seem equally important to me so it's a bit confusing.
The point of a Roguelike is that there isn't necessarily a winning strategy, you have to come up with it on the fly. Let me lay out the myriad ways one can die, and precautions against them:
"Projectile" laser weapons. Your bread and butter. Every ship should have at least one, if not more. These are prime shield-breakers, just on shot-count alone. No one strategy is going to be perfect against them. Shields will soak them up... ablatively. You will dodge some of them. A Defense II drone can shoot them down somehow. Your best bet, though, is to knock out his weapons room. Missiles, shields, and engines will serve you well, and having certain drones might protect you in passing.
Beam Weapons. These are great capitalizing weapons. Shields down? Hope you don't like 8 armor and some subsystems! The only direct counter to these weapons is shields, which they can only penetrate to some extent. Fortunately, the ones with scary shield penetration values tend to not show up in the first half of the game, anyway.
Missiles. A great opening weapon, softening shield generators and gun rooms for barrages based on more renewable resources. The only counters to this weapon are evade and defense drones. Good engines give you as much as 50% evade, and drones are 150 for the controller plus varying amounts for the kind of drone you're interested in. Drones are preferable, but engines tend to be easier and cheaper.
Bombs. You can't stop them, you can only hope they miss. High evade values, especially with Cloak, are your only defense besides swiss cheesing his weapons room, which you should do with great prejudice. Missiles are the the weapon of choice for these guys, and you should always keep a couple on hand, in case.
Teleporters. Not really a weapon, but they can mess your day up. Good airlock management, med-bay usage, and a spare crewman or two is the key here. If you can draw your enemies into the med-bay, they're basically done for. Have that upgraded, but only when you have spare change.
Fires. Have door upgrades researched, it's a pretty cheap buy. You should definitely have this by the end of the second zone.
Asteroids. Shields and evade. Mainly shields.
Asphyxiation. O2 controls.
As you can clearly see, there are a couple of consistencies:
Shields keep you alive
Engines keep you alive
Murdering the other ship keeps you alive
There are a few strategies you can try that seem to work for me:
Early investment in engines - anything that doesn't hit you is a return on investment. Follow that with investment in shields for maximum safety.
Cheaper, more-shot lasers are often more cost-effectiveness.
Invest a bit in everything, and a lot in a few things. This is called being "T shaped", and is considered a good standard in real life, too!
Teleporters are good if you can micromanage. Fortunately, you can pause the game whenever! Watch out for the automated ships, they're airless.
Sensors are more valuable than you'd think. If you've got spare change, consider them strongly.
Try and help the NPCs. Not every time you do will pay off, but it can pay off in a BIG way. Consider: Drone Recovery augment in the first stage with the Engi drone ship due to my going out of my way to help someone. It. Was. Glorious.
The message I hope you take away from this is that your question is, in a way, not the best one to ask. If this game was made well, there shouldn't be a real answer to your question. As it is, the best I can say is that some things have tended to work better for me, but at other times they've still failed spectacularly.
It's going to depend on a number of factors, for the sake of simplicity I'm going to ignore the possibility that you are trying for achievements.
What ship you're starting with is going to have a major effect on what you want to upgrade first. Also what gear you manage to find available either from salvage or stores will affect your choices as well. Ultimately every game is different, so there is no single always right answer, but here are some general guidelines:
Don't spend your salvage right away, especially early game. There is little more frustrating than arriving at a store that has the PERFECT item or crew member and you don't have the scrap to buy it.
Cloak and Teleporter can both be very effective if you can acquire them early. The latter is cheaper (75 rather than 150) but you'll need some extra crew.
Upgrade your Med bay, sensors, doors and O2 one point when you can spare the change. Even if you don't power them that makes them more resistant to damage.
If you can get your shields to level 2 (aka 4 bars) and powered early you can easily find a ship with no missiles that can't penetrate your defenses to train up your crew. Note that some people may feel that this is an exploit.
Don't neglect upgrading (and manning) your Engines. You can't rely on missed shots, but frequent misses can easily turn the tide.
Try different strategies. When possible I usually go for a boarding route (you can get a lot more scrap that way), but I've been effective with other means too.