>>528056Airports: Bucharest, Cluj, Sibiu are international airports either close to the Carpathians, or with good connections to them from the city itself. In the city, get to the train station and head for the mountains.
Easily reached from Bucharest is the Bucegi massif. Most popular hiking destination in the country. From Cluj or Sibiu, you can pick from a lot of options nearby, including but not limited to: Fagaras and Retezat - most truly alpine region - as well as the mountain ranges leading to them and connecting them. The Apuseni is a less mountainous alternative, with apparently a lot of traditional Romanian coleur locale.
Alternatively, head for Brasov, from which you have access to the area around Brasov itself, then Piatra Mare, Gârbova & Baiului and Ciucas. And more, probably.
Can make better recommendations if you specify, but I also suggest you try and find a copy of 'The Mountains of Romania' by James Roberts.