I'm making up a sentence from your words because context is always important to get a better understanding.
He drove our new car for the first time when we went to visit our parents.
Time here is a countable noun (countable means you can have it in the plural). The sentence means that at some point we acquired a new car and "he" drove it for the first time (for the first instance/occasion) in his life on the day we went to visit our parents.
After that there were several other times/occasions/instances on which he drove the car. And we could talk of a second time, a third time, etc. It's the meaning you mention in your comment: One of several instances.
In your original question you referred to the Big Bang and "the beginning of
the time" (no "the" here). Time in that case refers to "a nonspatial continuum" from past through present to future" (1.a). The beginning of time" is a phrase we might find in a religious books, but not only. It means the moment when our World was created. This is not referred to as "the first time, or "a first time", there is just one, it's unique.