If you want to travel to the Swiss sector, there are excellent train services to Montargis and some quite good local bus services from there into the Swiss sector. There is also a train service through from there to Chamonix at a limited series of times.
If you want to travel to Chamonix, the railway journey is complicated and needs several transfers between trains.
You may well find that the minibus transfers between Geneva Airport and Chamonix have much to recommend them. The transit time is about 75mins. Each of the companies have a published series of timetables showing many runs per day,but I don't know how many of these will actually run in practice. However, most people will be making a reservation for one of these times at the "shared transport" rate. A great beauty of the services is that they will pickup/deliver you to your hotel in the Chamonix Valley.
I have found four companies and I have given links to their websites below. My experience is very limited, but I can tell you the little I know.
Of the four, Alpy Bus is easily the cheapest but seems to be a cowboy outfit. A colleague and I booked with them and I gave them my colleague's (later) plane time for the booking. My colleague's flight was delayed just over an hour and there was no trace of Alpy Bus at the airport. We were abandoned totally! Even 90mins earlier Alptbus were not at the airport (I was there and looking for the vehicle). We found that one of the other companies had a bus at the airport and happily they took us to the destination.
I have never had an explanation from Alpy Bus of why their driver abandoned us. You might be surprised to hear that I persisted with the company on the return journey. They promised they would pick me up from where I was staying. In fact they transferred the booking to Mountain drop-offs who were there spot on time and even gave my colleague a lift into Chamonix free of charge.
About the same time the travel company for whom I lead tours used Alpy Bus to get some delayed clients to Chamonix. This was on my recommendation and I was very relieved that everything went to plan. Indeed a Google search shows several warm recommendations for Alpy Bus, but it also shows one local company who expressly warn their clients against booking with Alpy Bus!
Since then I have used Mountain drop-offs again and found them to be truly excellent. They are well-organised, helpful and on-time. They also send you a text message to your mobile the previous day which confirms the pick-up details and gives the driver's name and their mobile phone number.
I have possibly said enough, especially since I have only a very small number of events on which to judge. My own position is that I would mostly now pay the extra 60% that the other companies currently charge to be sure that I would not miss a flight. If I had a big cushion of spare time, then I might be tempted by the low prices of Alpy Bus. Otherwise I will always use Mountain drop-offs.
Airport Transfer Service
2006 TMB | TMB Info page