Taxis should be readily available at the airport. The ride to the center of Graz costs around Euro 25, maybe a little more at night and on Sundays.
You can also take the bus 630 or 631 ( timetable here) to town. Get out at the final stop "Jakominiplatz". To get to the conference location and to hotel Johannes or Mercure Messe from there, take streetcar 6 in the direction of "St. Peter"; For the conference and hotel Johannes, get off at "Neue Technik". For hotel Mercure Messe, continue for two more stops and get off at "St. Peter Friedhof". To get to hotel Mariahilf, take the streetcar 6 from Jakominiplatz in the other direction, "Laudongasse", and get off at "Südtirolerplatz". Here is a map of Jakominiplatz. The airport busses arrive at the lower left corner, marked "O", and streetcar 6 in the direction of "St. Peter" departs from position "C". Single tickets are sold on the bus. One ticket is valid for the whole trip using the airport bus and continuing with the steetcar. Cheaper tickets for 10 rides in the Graz public transportation network (busses, streetcars, airport bus) are sold in tabacconist's stores. The 10-ride tickets are called "Zehnerblock".
Another possiblity is to take a train from the airport to Graz Hauptbahnhof (main train station). Then proceed as indicated above "how to get to the conference from the train station".
The airport train station is about 300m away, straight ahead when you get out of the airport. You can take the baggage carts with you to the station. Airport train schedule here
Here is a public transportation map of Graz. The website of Styrian public transportation can be a bit confusing.