From Klagenfurt Airport to Slovenia
There are city buses (no. 40 and 45) from the airport to the main railway station in Klagenfurt between 6:00 and 23:00, departing every 30 minutes. The bus stop at the airport is located directly opposite the Arrivals entry hall.
From the main railway station there are several railway connections available to Villach from where you can continue your journey to Slovenia. For the timetable between Klagenfurt and Villach click HERE. The railway connections between Villach and Slovenia can be found HERE. 

From Graz Airport to Slovenia
The airport in Graz is a little more than 10 minutes drive from the main railway station, and the bus stop at the airport is located 300 metres away from the airport building. The timetable for the city bus can be seen HERE. From Graz to Slovenia (Maribor and Ljubljana) there are several daily railway connections available which can be found HERE.