Before we continue, it’s worth noting that one round can include multiple Genesis events.
You don't have to perform 100 Expeditions before your first Genesis initiation.
I suggest going on every expedition because I'm pretty sure that sending crew isn't counted in the total.
I believe that going on raids to ship wrecks and Framen/Mechanic ships counts towards the total as there was about 25 missed expeditions from my planet count that I can only put down to this factor.