The "that" you're interested in is a relative pronoun (corresponding to "que" or "o qual" in Portuguese).

The sentences you gave as examples are perfectly correct, but note that "that" can optionally be left out (and generally is in natural speech), unless it represents the subject of the relative clause.

So, you could also say "That's something I want to learn", leaving out the "that" which is the object of the relative clause "I want to learn [it]"; but in "I saw something that moved me" you must keep "that" because it is the subject of the relative clause "[it] moved me".

That is the day I was born. That is my birthday.
That is what I would prefer to talk about.
That is not a probative conclusion.
That is not logical.

That is not factual either.
That is not demonstrative of syllogistic reasoning.
That is not a Deductive Proof.
That certainly is enlightening.
If you will permit me, I will articulate that problem in this manner.
That question corresponds to the perennial Mind/Body Problem in Philosophy.

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That is not what I ordered.

That is not what I want.
That I want you for my wife, is what I am trying to tell you.
That is why I love you so much.

That you do not mind your own business is obviated to everyone.
That is not five pieces of shrimp.

That is four pieces of shrimp.
That is not what I ordered.

There is something wrong with that waiter.
That is one person I do not want to deal with right now.

That is not the worst thing you could have said.

That is actually not so bad.

That is absolutely the worst thing you could discuss

with a student in the first chat session.

Well, did you ever consider taking English lessons from that "Su.Ki" teacher?
"Su.Ki." is that teacher of English that has the painting of a woman or something in place of a photo.

That is about the worst description of a teacher on Italki that I have ever seen.
That you lack the facility of description, Bruce, is apparent to all.
I hardly think that is fair for you to say.
That is not the kind of shortcoming that really matters here.
That does not matter at all.
Bruce, that is the most comically absurd thing you could write about here.

That serves to show why you should not be discussing English with anyone.
Why? Do I look that stupid?
Exactly! That is exactly the reason.
That is a rather cruel thing to say, don't you agree.
I would never agree to that.
You know that I am right about that.
You cannot deny that.
That is possibly why we cannot be friends.
We cannot agree about that, or about anything else, either!

That settles the issue then, does it not?

It is entirely clear that I do not want to write any more about that.

That is not even something that I would worry about.
Why should I worry about that? That does not matter.
That it is an issue of consequence, is understandable.

That it is financially problematic, I can see; but what I do not understand is your
thought that financial ruin is my greatest concern.

That my family is safe is all that I care about right now.
That my friend, is one of the most poisonous snakes in the world.
That you should avoid that snake goes without saying.
That is not the subject here. The problem for that snake may be that he should be concerned that I do not bite his head off.

That kind of casual attitude could get you killed my friend.
That is why we came here, remember?
We came here to study that specific species of snake.

That does not mean we can justify reckless behavior or irresponsible behavior.
You might consider taking English lessons from that "Peachey" fellow.
Why would I consider taking English lessons from that "Peachey" fellow?

That is not all that is objectionable about you. There are other faults that I could talk about.
That your personality sucks, is obvious.

That you are completely illogical and lack critical thinking skills shows in that comment.
There is something that I want you to know.

I have wanted to tell you for the longest time that I care very much for you.

That is the most embarrassing thing that I have ever done.

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That I acted foolishly, I can freely admit;

but what really hurt was that you laughed at me.

That my friend, is the kind of behavior that is going to lead to somebody,

at some time, giving you a severe beating.

That you are "Asking---For---It" is obvious.

That you deserve a beating for talking like that to people

is apparent to everyone but you.