Learn Turkish: Ben bugün çalışıyorum.

The verb root is çalış- (work). We add the present continuous tense suffix -ıyor to mean "is working". Finally, we add the personal subject suffix -um (I), making it çalışıyorum (I am working).

Since çalışıyorum already means "I am working", do I need to include ben at the beginning?

Not always! In Turkish, the -um at the end of the verb already tells us the subject is "I". Including ben (I) is optional and usually just adds a little emphasis. You can completely drop it and just say Bugün çalışıyorum.

Why does the tense suffix use an ı (-ıyor) instead of an i (-iyor)?

This is because of 4-way vowel harmony. The last vowel in the verb root çalış- is ı. The harmony rules say that if the last vowel is a or ı, the suffix must also use ı.

Is the word bugün (today) made of two different words?

Yes! It is a combination of bu (this) and gün (day). So literally, it means "this day".