A true-story non-islamic example
About a week ago, news spread about "Samsung giving a billion dollars to apple as 5 cents!", even famous news sites like Yahoo! reported this legendary story, but was it correct? Actually No!.
Where did the error happen? What's wrong?
As you see a rumor spread like a plague which affected famous companies. This is because they didn't (apparantly) try to check the authenticity of it, although they should be masters of authenticity checking. This is only because the story was funny and people wanted it to be real (although I never believed it, I liked it very much and wished it be true :D ). So the error happened in the authenticity-checking step, which is the most important, but also likely to be most difficult, and hence comes the question:
Should I always check the authenticity of what I hear/read?
You should do that if you will ever repeat it. Because if it was proven to be wrong, then the one who heard it from you will have a bad image about you (as you maybe will towards the one who told it to you/the source you read it from). This is one reason.
Another is that you should check the authenticity of it if you know you are affected -willingly or unwillingly- by things you hear from wherever (which is a bad thing indeed), so you can't leave anything "hanged" in your mind.
A third reason is that you should check it's authenticity if you you thought it is not correct, this is because you are likely to just say "This story is not correct" without any proofs, and that will result in a bad image about you as well, disbelieving without a reason is sometimes a bad behaviour, besides, this story is taking part of your mind, why not free it and get some authenticity check experience? maybe with an idea about how authentic the source is.
But this is very hard, authenticity check is not an easy process
True, this is hard especially for, for example, bottleneck historical stories. Based on this, you should either just keep silent and not repeat the story, which is something most of us can't do! Or reference this story to the source where you got it (and maybe saying "I'm not sure about it's authenticity"), thus giving the listener a "handler" and maybe freeing yourself from the accusation of lying.
So, what does Islam say about that?
Islam seems to prohibit us from saying what we are not certain about, or tell stories that we author and are fake, as the ayah says:
وَلَا تَقْفُ مَا لَيْسَ لَكَ بِهِ عِلْمٌ ۚ إِنَّ السَّمْعَ وَالْبَصَرَ
وَالْفُؤَادَ كُلُّ أُولَٰئِكَ كَانَ عَنْهُ مَسْئُولًا
And follow not (O man i.e., say not, or do not or witness not, etc.)
that of which you have no knowledge (e.g. one's saying: "I have seen,"
while in fact he has not seen, or "I have heard," while he has not
heard). Verily! The hearing, and the sight, and the heart, of each of
those you will be questioned (by Allah).
But since authenticity check in Islam is a more complex process, which the majority of people can't do (neither do I generally), then it largely depends on the authenticity of the source. There is a Hadith which I think is very useful for this topic, it says:
The Prophet said, "Convey (my teachings) to the people even if it were a single sentence, and tell others the stories of Bani Israel (which have been taught to you), for it is not sinful to do so. And whoever tells a lie on me intentionally, will surely take his place in the (Hell) Fire."
حَدَّثَنَا أَبُو عَاصِمٍ الضَّحَّاكُ بْنُ مَخْلَدٍ، أَخْبَرَنَا الأَوْزَاعِيُّ، حَدَّثَنَا حَسَّانُ بْنُ عَطِيَّةَ، عَنْ أَبِي كَبْشَةَ، عَنْ عَبْدِ اللَّهِ بْنِ عَمْرٍو، أَنَّ النَّبِيَّ صلى الله عليه وسلم قَالَ " بَلِّغُوا عَنِّي وَلَوْ آيَةً، وَحَدِّثُوا عَنْ بَنِي إِسْرَائِيلَ وَلاَ حَرَجَ، وَمَنْ كَذَبَ عَلَىَّ مُتَعَمِّدًا فَلْيَتَبَوَّأْ مَقْعَدَهُ مِنَ النَّارِ ".
See there are three ideas we can extract:
The Prophet, and the Quran, are most trusted sources, the Prophet ordered us to repeat what they say because they are authentic, rule #1: If the source is ultimately authentic (and you understand it well), then repeat the story, advice... etc.
The Prophet allowed us to adopt israilites, but in caution. If the story contradicts to Islam then we shouldn't accept it, rule #2: Use the powerful brain with which Allah made you superior to all other creations. And check authenticity.
The prophet warned us from talking lies and refer them to him on purpose and said who does that will be in Hell. But what about the non-purpose, for example, saying a Hadith we think is Sahih but in truth it's not? This is the bottleneck, and depends largely on the reference, if you can't check authenticity, then look for the most trustworthy source at least, if you don't trust the source, then don't repeat what it says.
Hope that's helpful.