Some good breaking-news-viral-meme kinds of research tools listed here (Google's image lookup is hugely useful).
If it's not brand-brand-brand-new, some additional ideas I've found ueful:
1. As mentioned in the comments of the story below … look at the comments. People may already be debunking (or explaining or discussing) the origins of the tale. It may be hard to find them amidst the squawking, but it can sometimes be a helpful source.
2. Dig through to primary sources. If you're reading a story on an opinion site, drill down through the links to the news sources behind it. There may be additional information that was lost in the opinion "spin". (If the site doesn't offer links to news sources … consider that a warning flag as well.)
If it's just a visual, try just a plain old Google of the particulars (some text from the image, or some good descriptive words). There may be additional information already out there that can improve understanding of what's being shown.
3. If it's been more than a few days, http://snopes.com is, of course, an excellent resource (particularly for memes that keep rising from their graves to eat the brains of the living). For political items, a variety of political fact-checking sites exist — they are useful, at the very least, for reviewing the information behind things like some outrageous (or outrageously edited) quotation by That Candidate You Really Dislike.
As to what you should do when you uncover something as a hoax or egregiously unfair in how it's been edited and presented … it's sometimes tempting to get into a slap-fight with someone posting garbage ("You people who hate My Candidate are all evil hacks!"). It's also tempting just to let it drop (nobody will believe it, right?). I try (though I don't always succeed) to respond to the post / email / share with the facts of the matter, avoiding the name calling, to let people draw their own conclusions.
It might seem like even just correcting someone is setting up an argument, but the facts are important; without being reality-based, we can't do much to deal with reality. And you'd be surprised how many thanks you will get from people who honestly weren't sure, didn't know, had some doubts, and who really appreciate you doing a little research.
As Jefferson wrote: "Truth is great and will prevail if left to herself. She is the proper and sufficient antagonist to error, and has nothing to fear from conflict, unless by human interposition disarmed of her natural weapons, free argument and debate, errors ceasing to be dangerous when it is permitted freely to contradict them." [http://wist.info/jefferson-thomas/2082/]
And, of course, I left off some other ideas:
1. Does it even make any sense. No, really, does it? Or is it simply twanging the emotional / fear-driven response? That, at least, should be a warning flag.
2. If there's a citation, or a group taking responsibility for the story, is it from a hyper-partisan source (hashtags like "HillarySucks" or "RubioIsATurd" are probably clues about the intellectual rigor behind the article/meme, even if you agree with the sentiment).
As for responses, you're extremely unlikely to change the mind of the person who posted it, as that person has an emotional attachment to the idea, but you may well affect those who are reading it.
+John Bump Well, it's unlikely, but sometimes (speaking from my own personal experience) it's possible to effect a change, or illumination, for the original poster.
But, yes, it's more likely to be of assistance to others who are reading and going, "Whoa, he really said that?" or "That's outrageous, why haven't I heard of this before?"
Of course, there's also a certain amount of confirmation bias that will go on — "I believe this specifically is true because I believe this generally is true," which cuts both ways.
Still, better to light a candle then curse the darkness, and silence too often implies consent. I'd also throw out that Edmund Burke "evil to triumph" quote, but it's inauthentic.