Aggtelek together with Jósvafő are famous for their huge and beautiful karst cave system.
There is a direct bus from Miskolc which takes about two hours. It leaves at 9:45AM everyday day and a ticket costs HUF1,100. You need to change at Kazinc Bareika (HUF460) on the return as there are no direct buses. There is a bus every 30 minutes.
Hitch hiking is also a good option since it is probably going to take you half the time as by bus. The best bet is road 26 towards Kazinc first and then either take the bus or continue to Miskolc.