Learn English – I deposited the funds

I was the first to reach school today.

The idiomatic phrase is to be the first (person) to (do something) where the first determines person which is omitted. however, in spoken language, "the" is omitted most of the time.

I agree that "to reach school" implies that the person studies or works at that school, but I would add that to reach the school, would need a relative clause particularising "the school":

I was the first to reach the school from the town centre today.

Or

Today, I was the first to reach the school where the author of my favourite book teaches.