Airport: The Graz Airport provides connections with many countries in Europe and outside Europe. Many airlines transport every day an uncountable amount of tourists from all round the world to Graz. Among the companies traveling to Graz there are, for example, Austrian Airlines, Croatia Airlines, Air Berlin and Lufthansa. For more information visit http://www.flughafen-graz.at/ flights
To get the city center from the airport in an easier, convenient and fast option you need to go by taxi. It has a fee of 20€.
You will pay much less for transportation traveling by bus, that is 2.10 € for travelling to the city center.
Train: By train you can travel easily to Graz throughout Austria. There are also direct connections to Graz from Germany, Switzerland, Croatia and Slovenia. For this you can visit the main station in Graz - the Hauptbahnhof.
Bus: Other connections to Graz from all around Europe are, in instance, the bus company Eurolines. In many cases it is the cheapest way to travel long distances but it is not very advisable, since the trip takes a lot of time and is also tiring.
Car: The car can be useful especially when you have to scroll through more cities and go with lots of luggage. Without problem you can get to Graz and then travel throughout Austria by car as there are highways practically from all over Europe.