There are two daily buses from "stadionok" bus terminal (caution! it's NOT the central bus terminal at Népliget). You can get to the terminal either by metro (red line) or with tram No1, The daily departures are 6 15 and 7 30. Arrive earlier as you have to buy the ticket beforehand in the ticket office or in a vending machine. The terminal is not that big, there is a signboard that shows the nest departures - you bus will go to JÓSVAFŐ, there you will see which stand it departs from. The buses arrive at 10 43 and 11 18. Get off at "Aggtelek Kossuth utca, Barlang bejárati út (Kossuth str -way to the cave). This is the last stop before the Aggtelek bus terminal. From there take Kossuth str, (I'm quite sure that there will be some kind of signboard.). The entrance is cca 900 m from the bus stop. If you go in September take the 7 30 bus and you can reach the 12 00 tour. From October 1st the tour schedule changes, so you should take the earlier bus and you can reach the 11 30 tour. This is the short, 1 hour tour. Alternatively you can try to go - with the same bus - to Jósvafő and get off at the Szálló "Hotel" stop. As the entrance here is very close to the bus stop you have (LOW) chances to reach the 11 00 tour in September. Don't do that in October, as there is only one tour from Jósvafő entrance at 16 00,.
There are two buses back also 14 37 and 16 30
All other sites are in Slovakia, and they are not doable on a daytrip from Budapest by public transport.