Create the Best Circumstances for Learning
You should look to create the best circumstances you can for the other person being open to hearing what you have to say. People are more open to learning (which is by its nature uncomfortable and can feel destabilizing or risky) when they feel safe. So avoid starting a conversation in a location or situation where they might feel vulnerable. This of course depends on the person, but in general: private e-mails feel safer than public Facebook conversations. A one-on-one conversation on the couch with a glass of wine feels safer than in the middle of a family dinner party. A discussion that evolved organically out of a person’s comment or question feels safer than a forced lecture they weren’t ready for.
Obviously this is not always possible, but if you can, save the convo for in person or take it offline as quickly as possible. In person conversations make it easier to connect - you can see and respond to the person’s tone and body language, and you can better show that you care about where they’re coming from in person.
If you are online, perhaps in a comment thread on Facebook or Twitter, take your conversation to a private message or e-mail. Again, in “public,” people (including you) are more likely to be defensive and thus less likely to listen and learn. Also, avoid sarcasm and jokes - they just don’t always translate online and they end up confusing things.
That being said - standing up and speaking out right away when a friend or family member says something offensive is a vital part of fighting for justice. In those cases, do so firmly but kindly, then follow up later (see “Call Out the Behavior, Not the Person”).