In your second sentence:
I do take breaks
the "do" is redundant. This statement on its own would not be idiomatic English. It would be acceptable only if the "do" is serving a specific function. This could be (as Maulik suggests) contrast:
I don't eat grasshoppers, but I do eat rattlesnake.
It could be a response to a question:
Yes, I do like gladiator movies. Why do you ask?
It could be to draw some other kind of relationship to a previous statement:
We could go to Ljubljana for the festival. After all, I do have plenty of vacation days left this year.
The common thread is that there needs to be some previous statement that the "do" statement is referring to. This is common enough that if you just say the statement you suggested:
I do take breaks.
an English speaker will assume that you are responding to someone saying:
Do you take breaks?
You don't take breaks.
It would sound very odd to use this construction if there was not such a predicate.