jacob,
There are a number of Shuttle services that provide transportation from Geneva to Chamonix. They typically pick-up either from the airport or the main rail station. I believe I used Chamonix Valley Transfers on my last trip, and I'd highly recommend them, as they were very professional and dropped me right at the door to my hotel.
You will have to reserve seats on the Shuttles in both directions. They normally use large vans which have a limited capacity, so they need to know how many passengers they will have on each trip.
If you were planning to take the Mont Blanc Panoramic cable car across to Pointe Helbronner, I believe it's closed for maintenance this summer.
One other caveat to mention is that the Aguille du Midi is at an elevation of about 12,605 ft / 3842 M. If any in your party have altitude-related medical issues, you'll need to keep that in mind. If anyone starts to have difficulties, you'll have to head back to the valley ASAP.