I've played through several times on Normal, and after about three sectors I invariably find that I'm utterly outclassed by every enemy ship I encounter - my weapons can't do more than tickle them before they batter down my shields and blow me up, no matter what I do. How do I keep up with the game's power curve?
It helps to start out on Easy, where you get more scrap and ammo than normal. The only downside is a score hit at the end, but if you're having trouble surviving, score is probably the least of your concerns.
Don't rush things - if you've got time to explore, do so. Don't get caught by the rebels, but don't make a beeline for the jump point, either. This takes practice, but try to make Z or N patterns between nodes as you move closer to the jump point, so that you can hit more as you're on the run. The rebel fleet "resets" in each system, so making more or less jumps in a particular system does not change how fast they will catch up to you in the next.
The other thing that is important is to try to conserve your missiles - you shouldn't be using missile weapons at all in the first system or two, if you can avoid it. Obviously, if it's a choice between death or missiles, use them. Missile weapons pierce shields, so while you can take down shields and do damage with your laser weapons, you should. Shields regenerate quickly, so keep this in mind if you've got multiple laser weapons - you may have to avoid the autofire and target them in volleys to break through shields and do damage.
Try to stockpile scrap for shopping early on if possible. Sometimes your only chance at a really good multi-shot laser weapon will come early on, and this can make or break your game. Beam weapons are far more difficult to use, since they tend to require precision timing - they do the best damage when enemy shields are down. While shopping, make sure you've got sufficient fuel - I tend to like to have about 20 units on hand if possible.
On a similar note, if you can capture ships intact (by killing their crew) you can get the best rewards, including a better chance at new weapons. Destroying oxygen subsystems is a good way to do this early on, although weapons systems tend to be priority targets.
Also pay attention to your crew - keep your crewmembers at the same post and they will gain experience for working that system, which can mean big bonuses. Also, try to protect experienced crewmembers, especially your weapons officer. A maxed out weapons officer is worth his or her weight in scrap! I tend to assign the first person in the list as the pilot, the second as the weapons officer, and the third as the shield operator. That way, I can tell easily if everyone's in position.
Besides getting lucky and finding better weapons, you've also got several alternative options when it comes to combat:
It's hard to do more than one or two of these strategies at once, because of limited scrap for upgrades, and because it's just hard to manage drones, precision time weapons, keep tabs on your away team, etc, at the same time.
As far as upgrades go, I tend to prefer going for 2 shield units as soon as possible. Then I will focus on weapon system power, and/or system power for one of the optional systems if you've found it. After that, I focus on shields again. Many of the subsystem upgrades (ie, doors, sensors, medbay, oxygen) have less of a benefit compared to extra shields, so keep these at a low priority.
Sometimes, you're just going to do poorly even though you tried your best. Sadly, that's the downside of a "roguelike" type game - sometimes the dice are against you. Don't get discouraged!
Explore as many nodes as possible
You have to make sure you hit as many nodes as possible in the early systems before leaving them. Ideally, you don't go to the exit for a system until the fleet is right on your heels. In the early systems you want to take on as many encounters as possible, as most of the combats can be won with little to no damage, and you really want to stock up scrap before you start getting to the later, harder systems. Hit as many nebulas as you can, as they slow down the pursuing fleet and give you the opportunity to do even more encounters per system. Yes, it's riskier, but it seems better to me to take the gambles as early in the game as possible, when you have less time invested if the gamble goes bad, and when you have more time in the game to maximize your payoffs when it goes well.
Try to take enemy ships intact
Don't be afraid to let ships surrender to you. You often get as much if not more salvage from a ship that surrenders than one you blow up. In particular, attack any slavers that you see and try to get them to surrender to you so that you can pick up some extra crewman. Engis are very valuable for their repair speed as well as the number of encounters they can make easier for you, and mantises are great in combat. If you've got a teleporter, you can actually save a slave once you've killed the entire slaver crew, which is the best of both worlds, since you still get salvage.
Ideally you want to acquire a teleporter so that you can board ships and kill the crew. If you're lucky, you might be able to pull this off without a teleporter by targeting their O2 and suffocating them, but in my experience this only works in the rare instance that you manage to set fire in the O2 room and they can't repair it.
Save your scrap for stores
I also recommend you save your scrap for the first 2-3 systems and only spend it when you find something that really warrants it. Okay, sure, if you get 200+, go ahead and buy some shield upgrades or something, but it's good to keep about 150 in reserve for good items. Getting good items from stores early in the game pays off because it means you can bounce through a lot of easy encounters without taking much damage.
In particular, I keep a look out for the following:
I recommend holding off on the Cloaking Device until mid to late game. It's nice, but at 150 scrap, it's too high a cost unless you've completely given up on finding a teleporter and crew.
Target the enemy weapon systems
Another trick for the early systems (particularly 1 and 2) is to always target the enemy weapons first. Often you can disable their weapons in the first volley, and so long as you keep targeting them you can keep them disabled throughout the fight. It's slower than taking out their shields first, but it means you aren't getting attacked back, which in turn means you don't have to waste scrap on repairs.
You can probably get through the first 3 sectors with your starting laser. With 3 shots, it can plow through most shields and disable or at least damage enemy weapons in one burst. By the time enemies start having 2 shields, hopefully you've found a 2nd weapon to pair with it to help plow through shields. Even a weak Laser 1 paired with it means you can pump out 5 shots, enough to drop shields and usually disable a system if you time them to shoot together.This means you don't need to blow a bunch of scrap on weapon system upgrades. You'll probably want to get shields to 2 by late sector 3, but can typically get by fine before then simply by targeting enemy weapon systems as mentioned above. You don't need fancy door, sensor, or medbay upgrades this early, nor do you need engine upgrades. Those are all great later, but early in the game you really just want to do the minimal upgrades to make use of the gear you're lucky enough to buy/find, and always have a bit on hand if you find something nice in the store.
My personal strategy is to go for 2 shields asap. In the first three stages, most ships won't be even able to scratch your hull, meaning it saves a lot of scrap.
Second to that, pause the game before you do anything and investigate their weaponry. When I see misiles, I will use missiles (preferably Ion bomb) to take out their missiles. When I see beams, I'll first go for shields, if his laser or ion weaponry will proably not pierce you shields.
Boarding is a hard game to learn, how many times I killed my own crew because I forgot to turn auto off. For boarding there are hull missiles, which when used carefully, can net you a lot of extra gear, which is also valuable just to sell.
Upgrade engines first and always have crew in each section that specializes. Less hits = less hull damage = less money spent on repair.
Next, upgrade shields.
Apparently nebula is good place to get scrap but I am not sure if that is only because it slows down the rebels. Haven't tested this theory enough yet.
This helps me with the power curve at least until the end stage.