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...