Formal:
* Inaugurally
* Initiatory
* Pioneeringly
* Primarily
* For the maiden time
* For the inaugural time
* For the opening time
Informal:
* For the first go
* For the first shot
* For the first crack at it
* For the first time ever
* For the first time in history
* For the first time in my life
Figurative:
* A fresh start
* A new beginning
* A maiden voyage
* A baptism of fire
Specific to a particular event:
* Making their debut (for a performance)
* Setting sail (for a journey)
* Taking their first steps (for a baby)
* Inaugurating (for a new institution or era)
Remember to choose the best alternative based on the context and the desired tone.