I second the recommendation for Brasov as a home base!
The old town is fantastic. I suggest you stay in the Schei District, beyond Poarta Schei, close to Mount Tampa. There are great walks along the old city walls and hiking up Mounta Tampa and into the wilderness literally steps from your door. Sinaia and Peles Castle are an easy day trip, as are Rasnov Fortress and Bran Castle. You can also reach Sighisoara and Sibiu within a couple hours and there are loads of fortified churches, like Viscri, Homorod, and Prejmer within easy striking distance. Plus more beautiful mountains in nearby Poina Brasov, Predeal, and Busteni for hiking, biking, vistas, etc. If you're going in the summer, consider also the long but scenic drive via the Transfagarasan highway.
Timisoara is a lovely city, full of 1989 Revolution history and some fantastic architecture. The city center is great and there is a lovely river walk. Plan to visit Corvin Castle in Hunedoara and the Lipova ruins. If you choose Timisoara as one of your bases, you could consider long day trips to parts of Serbia (Novi Sad?) and Hungary (Szeged?)...I haven't been to Novi Sad and have only driven on the M-43 freeway near Szeged, so I can't really give you any details there.
Two other below-radar cities for a week-long stay in Romania are: Iasi and Cluj-Napoca. Like Timisoara, both are university towns with fantastic culture and architecture.
Iasi has a gorgeous Palace of Culture and a great atmosphere. There's also a cool monastery in town. Day trips to the Painted Monasteries near Gura Humorului and to beautiful Piatra Neamt. Even consider a stop-over in Chisinau, Moldova.
Cluj-Napoca is the most liveable and westernized Romanian city, in my opinion. Access to the Turda salt mines and a day-trip to the citadel and former capital of Alba Iulia. Long day trip to the Apuseni mountains. Cluj can also be used as a springboard to the Wooden Churches in Maramures (for a very long, but worthwhile day trip).
Drum bun!