Miskolc doesn't have an international airport, the nearest ones are in Košice Slovakia, Budapest and Debrecen.
By bus
Since Miskolc is well connected to the rest of the country by railway lines, there are fewer bus lines connecting Miskolc to other cities. See this page: (http://www.menetrendek.hu...) on information on buses. Unfortunately it's available in Hungarian only. Here are the words you need to know to use it: "honnan": from enter the city where you are, "hova": to enter destination, "keresztül": through, "mikor": when year-month-day, "keresés": search click on it after entering parameters. Now you get a page detailing time of departure, time of arrival and length of journey.
"naponta": every day, "munkanapokon": on workdays MondayâFriday, "Munkaszüneti napok kivételével naponta": every day except Sundays.
You can also take the bus no. 2 which goes from Búza tér large bus station in the centre of Miskolc to Tapolca. You can buy bus tickets on the station or on the bus for a somewhat bigger price. Bus tickets are 250 HUF or 350 if you buy it on the bus, ten tickets are 2000 HUF.
By train
The main railway station is Tiszai Station Tiszai pályaudvar in Hungarian. The trains going through Northern Hungary all stop in Miskolc, the city can be reached by train from all the major cities although if you come from the Transdanubian region, you might have to change trains in Budapest which means you have to find your way through the city to the Keleti station.
See the website of Hungary State Railways (http://elvira.mav-start.h...) for information on trains, travel time and prices. The site is available in Hungarian, English and German this link leads to the English version.
You should try to take the Intercity trains, they're much cleaner than regular and international trains, they don't stop at every small town and a seat is reserved for you. Intercity trains are a bit more expensive and it's better to reserve your seat days before the travel, because they are usually full.
The Tiszai station is well connected to the other parts of the city, both tram lines and many bus lines have their end station there.