A Priest’s literal first Mass is the Mass of his ordination, which he concelebrates with the ordaining Bishop.
Typically the next day the new Priest celebrates his first Mass by himself, which is also called a First Mass.
Then, for about a year a new Priest can celebrate what are called “first Masses” wherever he goes. These aren’t literal first Masses, although they would be the first Mass celebrated by that particular priest at that particular church. I believe he can also give first blessings for 1 year from the date of his ordination (you kiss the palms of the Priest’s hands after this blessing).
I can tell you for a fact that the first Masses in Phoenix were not literal first Masses, because I was present at the Ordinations in Winona as well as at all 7 first Masses of the new Priests in Winona this year. All of their actual first Masses were said there.