What is the most natural way to congratulate someone for having a baby. I know that 恭喜 means congratulations, but I think also 祝贺 means congratulations. Which one is better suited, or is there a better word I didn't mention?
恭喜 is more colloquial.
祝贺 is more literary.
Both can be used as a verb for "congratulate"
"Everyone goes [congratulating] Mr. Li who has added a son to his home."
Since 恭喜 is more colloquial, it is the one we use as an exclamation in speech:
"恭喜恭喜! 恭喜你們家中添了個小男孩"(Congratulations!! Congratulations on adding a son to your home)
"恭喜恭喜! 恭喜你們家中添了個千金" (Congratulations!! Congratulations on adding a daughter to your family)
恭喜恭喜! is the exclamation (optional, but very common)
恭喜你們家中添了個小男孩 (or 千金) is the detailed congratulations. This phrase can be used in speech or in writing
You should only use the literary 祝贺 in writing. And you don't need to add an exclamation
祝贺你們府上添了個小孩 (Congratulations on adding a child to your home)
Related term: 恭祝 (wish/ bless)