First thing or the first thing

In casual speech, yes, you could drop the definite article. Nevertheless, most Americans would probably use the definite article. The sentence is correctly written, although rather wordy. A more common way of saying this might be: The first thing I usually do in the morning is shower. "Early" is not really necessary unless the speaker wishes to emphasize that he or she is an early riser. The definite article is dispensed with in expressions which are arcane (i.e., one simply has to learn them). First thing, Bill, get the truck ready to go. First thing we do is knock some heads together. I'll be there first thing tomorrow. One would dispense with the definite article in your sentence if it were much more abbreviated: First thing I do in the morning is shower. Again, this is arcane, one justs has to have a sense of how the language is commonly used. Because of the rather long locution ". . . I usually do early in the morning . . ." the sense of our language tells me to use the definite article.

Is it also possible to write, "The first thing I usually do in the morning is TO shower"?

Thank you

In British English there is a set phrase, "first thing in the morning", which is an indication of time and means "very early in the morning". It can be shortened to "first thing".

I'll see you first thing in the morning.

I'll be there first thing.

Is it equally correct to say, "The first thing I usually do in the morning is TO shower" or is it better to drop the "to" in front of "shower"?

Yes, that is perfectly acceptable, and commonly used. There is no reason to prefer one form over the other.