Hmmm, by sectors 3 or 4, you should definitely run across something on the offensive side. This could be another weapon (and these can also be picked up outside of shops, if you are lucky), a teleporter, or a drone control / offensive done.

Even failing that, the starting ship has some decent weaponry; the missile can go through shields (so if you time it together with your lasers), you should be able to completely disable a 2-level shield with your first salvo. You may want to go for their weapons first, though. Disabling their pilot / engines removes their ability to dodge, meaning that all 3 of your lasers will hit.

I'd also recommend reading through the various wikis that exist, e.g. ftlwiki has great tips / strategies for each player ship. Here's their take on the Kestrel (starting ship):

http://ftlwiki.com/wiki/The_Kestrel

Other simple tips would be - always attack (and defeat) slavers, they usually offer a slave in exchange for mercy, which is a great deal. If they are non-human, you will get more "blue" options, which usually have (better) rewards. Later on, simply get to level 5 to unlock the engi ship, and play around with it to get an idea of how powerful drones can be.

In summary, the gameplay is actually very balanced, and pretty deep. With a lot of skill, you should be able to get to the final level on easy most of the time (and perhaps beat the boss half the time, depending on your strategy). But being a kind of roguelike (with all its randomness), sometimes you will just have bad luck...

Missiles penetrate shields. Target their shields system with your missiles and then target their weapons system with your lasers and you should make short work of most enemies. With the starting ship (The Kestrel), this strategy can get you through sector 4 without any significant upgrade.

To add to what others have said, one of the most important tactics is timing your weapon usage in combat.

For example, if you're using the Kestrel (Missile and Laser weapons), then you want to hit Shields with the missile first to disable them. Only if you actually downed the shields do you want to shoot the laser so that all three shots have a chance at doing damage (preferably on the Weapons subsystem). If everything hits (which happens very often in the first 4 sectors), then you've just disabled the enemy's shields AND weapons - they are now easy pickings.

In summary, never use autofire (unless you have Ion weapons) and time your shots so each weapon gets a maximum chance of doing damage.

Don't use autofire.

Another useful tip is to get those doors upgraded ASAP, then work on upgrading shields. Fully upgraded doors are lifesavers when fires break out or intruders beam in.

If you can't defeat a particular enemy you absolutely should not be fighting them. Hunker down, bring up your jump as quick as possible and get out of there. Then concentrate on building your ship up until you are more formidable.

The real trick to consistently beating this game is to always horde at least 75 scrap, and buy a Teleporter as soon as possible. Then, buy extra crew members. Preferably, these will be Mantis > Rock > Most Others > Engi & Zoltan. Then, near the start of every battle, fire a single volley to set things on fire / distract crews, and immediately send over your boarding party to begin harassing them. If there are too many, pick on them, or attack things while they try to put out fires. Often, this will cause them to give up on one repair, wasting time while they run around the ship. If they're a small crew, then just roll over them with Mantis / Rock strength. Just make sure you don't attack too much, or you may kill your boarders.

This is doubly effective, as if you kill all of a crew, then you (almost) always get the best reward possible, which often contains bonus equipment, and up to 50% more scrap.

By the end, you'll often have so much money that you'll be capped on energy, and just buying alternate max systems. Also, always destroy rebel ship Medbays, and the Fire Beam and Bio Beam are devastating for this strategy.

Finally, as noted in another person's comment, Teleporter makes the last boss remarkably easy. Just Teleport into each of the weapon rooms and kill the operator (starting with Missiles). Notably, don't kill every crew member, or the Super AI will take over for the final boss and it goes into crazy, Ultra-mode.