How to deal with today's world?

[PaperclipGames]

I'm not so much asking for an answer to that question specifically, but it is the center point of what I feel like I should address.

We can all agree that there is evil in this world, and that horrible things happen. We hear awful things on the news on a daily basis - overdramatized or not, but they still happen. When I sit in front of the telly and mum is sitting on the couch next to me, and some case about a man locking up people and torturing/raping them before killing them comes up, my mum would react in some sort of shocked way, clearly feeling disgusted about the fact that that happened. But it doesn't really affect me at all, because I don't know the people in question, and basically, in no way does it affect my life.

The question is, how rationally should one be in this world? I normally try to be as rational as possible: not let anything bad get to me, because what would that achieve? What if I felt disgusted by the news? I'd just feel worse, but at the same time I can't make a significant change, so in the end that'd only mean I've let myself feel bad while I also could just have stayed happy. So I simply "ignore" the bad feeling/thought and just go about my own business.
On the other hand, if someone is as passive as me about these emotions, they won't actively try and change the world, either, and the evil would remain. (I personally don't believe a single person, and at least not me, can make a big difference; but if we hypothetically said everyone were like me, then there would be no big demonstrations, or, for example, very influential and inspirational people to "save the world".

So, the dilemma is basically as follows.
- If you lets bad people influence you and make you feel bad, then it may (or may not) be possible for you to decide to stand up against them, and thus create a change and try to eradicate this evil, but in a lot of cases you'll only end up feeling bad without making a difference. You'll be a vulnerable person. So this might be bad for you personally, but it allows humanity to improve and solve big issues.
- If you don't let bad people influence you whatsoever, and/or simply block whatever negative that comes down your path, you're a lot less vulnerable and in a way a stronger person, but you're likely to have a more passive stance towards everything (because you don't have strong enough feelings about these bad things to really actively do something about it), so while this may be the better solution for you personally, it won't help actually solve anything.

I am a guy who has been bullied for the greatest majority of my life by my peers, been dragged down into serious and suicidal phases, and I figure that, considering I was never built to ever be a strong person, the main defense mechanism I could build was to simply stop attaching any value to what they said and did to me; and ignore them (in case of verbal things) to the best of my abilities. In a way I view this as a strength, as I am capable of not letting anyone get to me; but at the same time I've found it to be a weakness (personally), by being almost completely unable to really trust anyone or develop positive feelings. I have selectively let go of this invulnerability, and feel comfortable with this.

How do you guys cope with all the bad in the world in general, and do you think that one should be open for influence/improvement or rather not?