When can you omit "that"?
There are a number of instances in English where it is possible and even desirable to omit that.
that as conjunction with reporting verbs (like learn, discover, find out, know, feel, etc.): I discovered (that) Julian had borrowed my car without my permission.
after the more common reporting verbs, (e.g. say, tell) it is also entirely natural to omit that in informal speech: I told him (that) I'd be back by ten o'clock but he said he needed me here by nine.
after certain verbs (e.g. reply, shout) that cannot be omitted and it is not normally dropped after nouns: The Dean of the Humanities Faculty informed the students that the drama dept was going to close.