It will be the first time I have flown to America - why not just present simple?

"It will be the first time I have flown to America." - the flight will have finished at the point in time you're talking about: "It will be the first time [that] I [will] have flown to America". The first part of the clause indicates a future time, the subclause "i have flown to America" is in future perfect, indicating a completed future event.

"It will be the first time I fly to America" - you are talking about a future event from the perspective of the present.

The present simple tends not to get much usage in English as it does in other Latin-based languages. Typically it's used for describing an action that happens at an unspecified, or multiple times:

I fly to America every month or I fly to America if my work requires.

Rather than the present participle for when you're actually doing the thing: I am flying to America.

Since your example is talking about a specific, single time, it's perfectly reasonable to say It will be my first time flying to America. but it's not really an easier construction than It will be the first time I have flown to America.