Hello
@cathyke50
Simple answer is that the connection from Aosta by bus is cheap and quick. I took it on a Sunday morning in July 2018 and it left about 10am
The bus station in Aosta is near the main street and you can just buy your ticket at the station.
I haven't done it from Geneva, but it is a much longer distance. And a small point perhaps, but inevitably more expensive to travel in Switzerland than Italy, but you may have other considerations as well. I have no experience of Aosta airport. There are good bus connections and train too I think between both Milan and Torino airports and Aosta.
Late June the road to GSB is guaranteed(!) to be open. You might find some useful information and further discussion here
https://www.caminodesantiago.me/com...-to-crossing-the-great-st-bernard-pass.59812/
The Confraternity of Pilgrims to Rome is also a useful source of information.
https://pilgrimstorome.org.uk/
You might consider joining and you will get your credencial as part of the package!
Disclaimer: I am a trustee of this entirely voluntary organisation.
Best of luck with your plans. Let's hope things will open up fully this summer.
Oh and you will see my suggestion (with which not everyone agrees) that you could consider getting the bus from Aosta back to Bourg St Bernard and walking
up to the hospice as well - you could even leave your bags at the hospice as you pass through. You have to change buses in the summer anyway if you wished to continue 'backwards' into Switzerland. It is a bit of a climb, but at that stage really a recreational one. Then stay at night at the hospice and start down the next day. In busy times, it is worth booking ahead at the hospice. They can get full with groups of children in the summer. There is a very expensive hotel on the Swiss side as well - which belongs to the hospice. And a much cheaper hotel on the Italian side, just 500m around the lake. But the hospice is an iconic place to stay if you can.
Buon Cammino.
Tim