Arriving by plane
The closest airport is Graz Thalerhof. From the airport, you can take the train line S5 to the main railway station or a bus to Jakominiplatz (the central hub for public transport in Graz).
You can also take a train from Vienna International Airport to Graz.
Public transport in Graz
Graz has an efficient public transport system based on buses and trams. Perhaps the easiest way to purchase a ticket is to use one of the machines within the trams or at some of the bigger stations (in particular, at the main railway station and Jakominiplatz). Moreover, Hotel Stoiser will keep some week tickets ready for you, so that you can buy them directly at the reception desk of the hotel. You can also purchase tickets at a Trafik (kiosk). Hour and day tickets can also be purchased from a bus driver (but not week tickets!).
Possible choices for your accomodation: