To misunderstand or misinterpret something, especially in such a way as to take offense at what is said or done. Now, I hope you don't take this the wrong way, but I think it would be easier for everyone if you waited to visit until the baby is a few weeks old.

What is it called when you take something the wrong way?

Misunderstand, misinterpret, especially so as to take offense.

How do you say don't take this the wrong way?

A way to get around the clunky feeling of "don't take this the wrong way" would be to replace it with "please don't take offence" or something else more specific to the context of the sentence. Getting around using "don't get me wrong" is even easier.