Fact-checking seems to be an obvious response to fake news, and verifying the facts of a story is an invaluable service to everyone involved in a conversation. However, fakery often relies on a mix of facts, unfounded assertions based on facts and partisan opinion, and speculation. Constructive discourse calls for responses to the story as a whole, and the point of the report, as well as the facts.
That said, here are some places that can help you check the facts:
If you suspect that something is fake, you can do your own internet search and see what's behind the story. Use your favorite search engine or library service and do a search on some of the phrases that raise questions for you. Do the results come from familiar, mainstream news sources, or do they come from sites that all represent the same point-of-view? Do they pass along or echo the same story with no additional investigation? Can you find the original, primary source and verify its legitimacy?