The atheism Reddit has a page dedicated to this topic (https://www.reddit.com/r/atheism/wiki/comingout).
The atheism Reddit recommends NOT coming out. They are quite serious about it and go on for some length about it. Their basic idea is that if you are dependent on your parents, or if you are in a society which is violent towards atheists, you should be very careful before letting them know you are one.
The link above includes several dozen stories of atheists coming out to their families, and having awful things happen as a result.
My advice, if you come out, is a little different. I recommend applying the principles of Street Epistemology (https://streetepistemology.com/resources/).
A very great many people, especially those who are very emotionally attached to certain beliefs, DO NOT CARE how you get to a belief. Fear, pressure, thinking errors, whatever. The process doesn’t matter, only the result.
That means, people are much more comfortable discussing processes than they are questioning results (note: they still might not like discussing processes that much). They’re less attached to reasons than they are to conclusions.
Don’t make letting others in on your thoughts be some dramatic “coming out.” I recommend starting by talking about your thought processes, not about your conclusions.
Don’t say, “I have something to tell you; I’m an atheist,” but maybe just ask something like “Why do you believe in God, Dad?” Maybe explain why you are or aren’t persuaded by whatever they say, and then drop it (if you can).
Another time, ask things like “Why is faith a good thing, anyway?” “Are you a hundred percent certain? How can you be so sure?”
This may still be a little uncomfortable, but people are less uncomfortable talking about reasons than they are about conclusions.
If you’re asked what you believe, or if you’re an atheist, you might say something like “I think it’s possible that there’s a God. I think it’s possible that Jesus (or Allah (or Vishnu, etc.)) is the right God, even. I just can’t seem to find any reason that shows that He is for sure. I need a reason.
Eliezer Yudkowsky uses this line (for different things): “I’m willing to believe X is true, but I can’t see how we know it’s true.”
People can have allergic reactions to labels like “atheist” and “don’t believe.” Very, very often they think it means that you hold a certain, positive, unconditional belief that no God exists (and can never be persuaded otherwise), the same way that they hold such a belief that God does exist. Consider, if saying “atheist” makes them believe something that is not true about you (that you hold a positive belief), then the most honest and accurate way to describe your beliefs may require that you avoid labels, and just describe your thoughts and feelings (“I need a reason,” “It could be true, but how could I know if it was?”).
The important thing for honesty is not to say things that are true, but to try and make other people hear things that are true. A statement that says one thing, but is understood to mean something else has failed to communicate a truth. If a technically true statement is purposefully used to give others the wrong idea, then it’s as dishonest as an outright lie. More important than what you say is what they hear (as far as communication goes).
So, if saying “atheist” makes people think that you hold a positive belief that there is no God, then you may need to find another way to communicate what you really believe/don’t believe.
The principles of Street Epistemology are wonderful for having productive, friendly conversations, that get at the core of people’s thinking. Anthony Magnabosco often finds people shifting their confidence in god from 100% to 80% and lower in his SE conversations, and the other person walks away happy about the conversation, not offended or defensive. They’re willing to come back. They recommend their friends to talk to him.
Amazing stuff. This video is a great example of not making these delicate discussions into defensive arguments: