Arrival by plane.
If you arrive by plane, the easiest (and most comfortable) way is via Zürich Airport (ZRH)
In the basement of Zurich Airport you will find the railway station, which is also called "Zurich Airport" (German: Zürich Flughafen). There is no need to leave the Airport, just use the escalators and follow the signs.
From the railway station "Zurich Airport", there are direct railway connections to St. Gallen Main Station (e.g. using IR37, IC1 or IC5). Tickets are about CHF 28,00.- and Swiss railway is very reliable and on time :-)
You will find an overview of the connections from Zurich Airport to St Gallen and prices on the homepage of SBB (Swiss Federal Railways).
If you choose another Swiss airport (e.g. Geneva (GVA) or Berne (BRN)), we recommend to use the train as well. Arrival by train.
St. Gallen is very easy to access by train. For this purpose check-out the websites of your choice, (e.g. Deutsche Bahn
or Austrian Federal Railways