Learn English – “A first time” or “the first time”

This is the first time I've lost my passport.
The first time I went to London, I lost my passport.

The first time is a specific time, hence the definite article. “*A first time” would imply that there are many first times.

It is also possible to say my first time, although that is mostly done when there is no verb.

This is my first time losing my passport. [Possible but not very common]

This is the first time I've been to London.
This is my first time in London.

A first time is rare. It can be used in when referring to the concept of doing something for the first time, with no specific thing in mind.

There's a first time for everything.

When referring to a specific circumstance that is encountered for the first time, you can say a first (for me).

Losing my passport was a first for me.

First-time can also function as an adjective, to mean a person who is doing something for the first time: a first-time voter (someone who is voting or will vote for the first time in his life), a first-time writer (someone who is writing his first book), …