Well, early on when your engines are low, and your pilot and engineer inexperienced, you're not dodging too many shots, so it may seem like they're not worthy. I don't know that they're more important than anything else, as shields and weapons are important, too. (Boarding helps if your weapons are weak.)
It took me quite a bit of time before I got good. I forgot to check how many games/hours, but while I was often able to make if far early in my career, when I finally got to the boss, my ship was weak, and I got shredded. When I was finally able to limp past the first round, I got even more shredded in the second.
I got so frustrated that I temporarily gave up on beating the game, and just went on unlock missions. I unlocked nearly all the ships, including the ? cruiser, which I spent many frustrating hours hunting in the Rock A, as I was also trying to unlock the Rock B.
Eventually I watched videos to help me learn. Twinge did a
Kestrel tutorial, which is very long, but you may find it worthwhile. 5th Horseman made [urlhttp://www.ftlgame.com/forum/viewtopic.php?f=14&t=12033]a lot of vids [/url]which I enjoyed - it's funny to watch other people's mistakes (and props to him for posting them).
In the other thread you asked how I can max systems. Well, generally speaking I prioritize improving defenses first: see
this post by 5th Horseman. Less damage taken means less scrap spent on repairs, and thus more scrap for improving your ship. Also, don't repair your ship fully - going to 25 HP leaves room to take advantage of events where you hull gets repaired, or a repair station which has cheap repairs (and when you get better, repair to 20 HP).
If you start out using missiles, don't have it on autofire, and maybe accept more surrenders early, when they're offering lots of missiles (say 4+). Yes, you get more scrap when you blow up a ship, but early on it's not much more, and consider the scrap equivalent value of missiles, etc. if you were to buy in stores. And try not to buy missiles in stores. You'll want to switch to energy weapons, or at least bombs, which can't be shot down. I like, in ascending power: small bomb, breach bomb Mk II, ion bomb.
Did you see
this thread about dealing with Mantis boarders?
You get more scrap by killing the enemy crew rather than blowing up their ship. Boarding is the primary way, but anti-bio beam, fire beam/bomb, suffocation, or even judicious use of weapons can work.