Learn Romanian: Copilul crește din zi în zi. - The child is growing day by day.

Yes, the structure din [noun] în [noun] is a common Romanian pattern meaning from one unit of time to the next, or day by day. You can use it to emphasize gradual change or frequency with other nouns, such as din an în an (year by year) or din oră în oră (hour by hour).

The verb crește means to grow here. Can I also use it transitively, like to grow a plant or to raise a child?

Yes, a crește works both intransitively, as in this sentence where the child grows, and transitively. If you want to say you are raising a child or growing plants, you use the exact same verb: părinții cresc copilul (the parents are raising the child) or eu cresc roșii (I grow tomatoes).

Why doesn't zi have a definite article in the phrase din zi în zi?

In Romanian prepositional expressions of time or measurement that repeat the same noun, the noun takes the zero article, meaning no article at all, just like in the English phrase day by day. Adding a definite article here, such as din ziua în ziua, would be grammatically incorrect.

Could I say în fiecare zi instead of din zi în zi in this context?