Sibiu is a great town, with everything from a vibrant night life to the oldest museum in Europe. The middle-ages part of the city is wonderful, and if you visit it during August you might just be there for the medieval festival.
Ah, Sighisoara. Just an absolutely phenomenal place to be lazy, relax, and meet some interesting locals. The locals in Sighisoara (and Romania as a whole) were the nicest people I met in my life.
Brasov is just fantastic. A town with charm and much flair. The people are (like many Romanians) are very hospitable, nice and helpful. Explore Sinaia (Peles castle), Rasnov, the fortified churches of Prejmer and Harman. And if possible don't miss Sighisoara, the medieval town two hours from Brasov. It's great there …
I urge anyone is planning to go to Bucharest to stay well away. It’s damp, smelly, thieves, beggars, insects and bad sanitation (I mean real bad).