Learn Turkish: İlk gün çok zordu. - The first day was very hard.

Both ilk and birinci mean "first", but they have slightly different uses. İlk means "initial" or "the very first" (like the first day, first time, or first love). Birinci is the literal ordinal number "1st" formed with the -(I)ncI suffix (used for things like coming first in a race). For "the first day" of an experience, ilk gün is the most natural choice.

Would it be grammatically wrong to say birinci gün?

It is not grammatically wrong, but it changes the context. You would mostly use birinci gün if you were making a list or strictly counting a sequence of days, like "Day 1, Day 2..." (birinci gün, ikinci gün...). For talking about your initial experience (like the first day of school or a new job), you use ilk gün.

Why does the sentence end with zordu instead of just zor?

Because the sentence is in the past tense ("was very hard"). In Turkish, "was" isn't a separate word; it is the past tense suffix -DI attached directly to the end of the noun or adjective. Because the last vowel in zor is 'o', 4-way vowel harmony turns the suffix into -du, making it zordu.