Hi
We moved here with our two cats. We shipped them on their own from LA to NYC in cargo, to stay with relatives for a couple of months, then flew with them from NYC to Zurich via Atlanta with them in the cabin.

As others said, the Swiss are very cool about importing standard domestic pets, but they need to have their rabies shots and I think they need to be chipped (that's what we were asked about upon arrival, though they didn't ask us for proof.)

One thing I learned in this, in spite of anything else they may suck at, Delta airlines as far as I know is the most animal friendly american airline in terms of service, price, and regulation. I think they had some problems with animal shipping a few years ago, if that's teh case, they have clearly worked on it.

Oh and DON'T transit through the UK if you can; they have a slew of really awful regulations regarding animals.
Cheers
L.