You will get within about 30-45 minutes drivingtime from Buchs or Sargans to the town of Appenzell. These roads are in a very good condition, but quite narrow mostly, but there is almost no heavy trafic.
Once that you arrive in Appenzell you will find several large Park & Ride-areas, from where you can easily walk to the towncentre within 3-5 minutes. Parking there is free of charge !
There is a small side-trainline taking you to Appenzell and it has a narrow-gauge railtrack that will go mostly parallell to the mainroad to the town of Appenzell. This trainline is part of the SBB-trainsystem, so your Eurail- Interrail and similar prepaid tickets will be fully valid there as well.
Appenzell is just quite a small village and from the train-station you will get to the centre of the village within just 3-5 minutes walking-time.
The private, narrow-gauge Appenzeller Bahn (or "The Red Train" as Minifrosch called it) is the primary means of public transportation in the Appenzell area. Regular trains run from St. Gallen to Appenzell, as well as from Gossau to Wasserauen via Herisau and Appenzell. If you're coming from Zurich, you'll most likely change trains in Gossau. From Luzern or Lake Constance, the best connections are the Voralpen-Express to Herisau, changing to the local train there. The trains are also useful for getting around the area, most notably to Wasserauen for the Ebenalp lift and walking path to the Seealpsee, and Jakobsbad for the Kronberg lift and summer luge. If you stay in a local hotel for 3 nights or more, you will receive a Gold Appenzell Card, which gives you free, unlimited use of the railway during your stay (Note: Silver Appenzell Cards, for longer stays at guest apartments, do NOT provide the transport benefit). Otherwise, one-way local zone tickets cost CHF 6.40 in 2012 (Swisspass and various 50% discount cards valid).
One minor warning: Be aware these trains stop running rather early (around 7PM or so), so keep that in mind when making your plans to arrive or depart the Appenzell area.