Quoting BoomBoomWo
Geneva - Berlin
Berlin - Vienna
Vienna - Venice
Venice - Rome
Rome - Geneva
All in 10 days? Sorry, but are you crazy? That would be a 3 weeks trip, not a 10 days trip.
To answer your question:
Geneva-Berlin: easyjet flies this route for 29 EUR one-way
Berlin-Vienna: either plane or train. There are train specials available for 29 EUR one-way, but the train takes about 7 hours. IIRC Germanwings flies this route for 19 EUR.
Vienna-Venice: Train. There is a night train, specials are available from 19 EUR on.
Venice-Rome: Train. The Eurostar train take about 4 hours.
Rome-Geneva: Difficult. There are very few trains that go north from Italy into Switzerland. It might be cheaper/better to fly.
If you are flying in from the USA you might be better off flying into one city and departing from another. Say you fly into Munich, then go to Vienna or Berlin and fly out of Rome.