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.
48.083320.51Bükk (Bükk or Bükk hegység – literally Beech Mountains) (You can get in on road No. 25. from Eger. You can enter on the M3 Highway from Miskolc. Or, if you come from Slovakia, you can choose Road No. 27. and Road E96.). A section of the Carpathian Mountain range. Located in counties of Heves and Borsod-Abaúj-Zemplén. There are some interesting caves: Prehistoric Cave, Istállóskő Mountain; Szent Anna Cave, Lillafüred; Szent István Cave, Lillafüred. Do: You can tour on the paths in the forests.
47.918618.93562Börzsöny (Hungarian: Börzsöny, literally: Logwood). thanks for good location one of the most popular resort place for Budapesters
47.892419.42113Cserhát (Hungarian: Cserhát - literally "Turkey Oak Back", where "Cser" is Turkey Oak and "hát" is back). Includes Hollókő, an UNESCO World Heritage Site
48.164819.78144Karancs-Medves area (Slovakian: Mučínska vrchovina) (Near to Salgótarján).
47.898920.01885Matra (Mátra) (Located between the towns Gyöngyös and Eger. From Budapest turn off from main road at Gyöngyös and drive toward north. By public transport maybe the easiest way take a direct bus to Kékestető). Include Kékestető the highest peak of Hungary (1014m a.s.l.), a popular winter sports resort.
48.41121.40416Zemplén Mountains / Tokaj Mountains (Zempléni-hegység/Tokaji-hegység). Spotted with castles and ruins, a wine region that is also member of the UNESCO world heritage site list