BUCURESTI
The renovations and rebirth of the Old Town and city center are amazing! Great pedestrian areas, parks, shopping, restaurants, and nightlife. The buildings are now colorful and it seems there is green vegetation everywhere! Palatul Parlamentului (The Palace of the Parliament) is an immense building (beyond anything you can imagine) and absolutely worth a visit. There's a great city walk from Piata Unirii, by Curtea Veche and Palatul Voievodal, to Revolution Square - right through the heart of old town. The village museum near Parcul Herastrau is exceptional.
BRASOV
One of my favorite European cities. It wraps around the mountain and oozes that European city feel. Look for lodging in the Schei District. Perfect access to Mount Tampa and the hiking trails, the old city wall and bastions and easy walking distance to the Black Church, Council Square and Strada Republicii. Brasov is also a perfect base for day trips to visit Sinaia and Peles Castle, Bran Castle and Rasnov Fortress and the various fortified churches throughout the region. The fortified churches are easier to visit by car.
SIBIU
Corvin Castle in Hunedoara - about an hour west of Sibiu - is one of my favorite castles in Europe. Reminds me of Hogwarts. Sibiu is a fun city in itself and there's a great ASTRA open-air folk museum nearby. Great for a base to visit Corvin Castle and nearby Alba Iulia. Alba Iulia - of historical importance to Romania, Hungary, and Transylvania - has a fantastic fortress with huge bastions, Roman ruins and a surprising 11th century catholic cathedral.
SIGHISOARA
I would put Sighisoara on the must-see list too. Whether or not you're interested in Dracula, the city's medieval clock tower and old town are great. So is the wood-covered stairway up to the church-on-the-hill. You can make it a day-trip from Brasov or Sibiu if you have a car. It's difficult to get by train if you're coming directly from Sibiu. Otherwise, it's a great stop en route between Brasov and Cluj or the long way from Brasov to Sibiu.
CLUJ-NAPOCA and TIMISOARA
Cluj-Napoca and Timisoara are underrated. Both cities have a wonderful vibe. Timisoara has exceptional history, from the Hungarian days but especially with regards to the Revolution. The town is great for walking and has a lovely riverfront. It's also an easy 3.5-hour drive from Budapest to Timisoara. About 5.5-6 hours by train.
I feel like Cluj is the most westernized and the most livable city in Romania. It has a growing entrepreneurial culture and is a hotbed for tech companies. They call it the Silicon Valley of Transylvania. It has great, historic churches and provides great walks through the old town and city center to the river front and towards Fortress Hill. The Turda salt mines are an easy day-trip from Cluj.
MOLDOVA (region of Romania, not the country)
Moldova is more difficult to reach, but worth a visit...usually on a subsequent trip. The painted-monasteries draw the tourists - and rightfully so - but Iasi has a majestic palace and I love Piatra Neamt! Just the feel of the city and the access to nature. Love it. FWIW, the drive from Cluj to Piatra Neamt via Bistrita and Vatra Dornei is one of the finest around. It includes the Borgo Pass (from Braham Stoker's Dracula) and Hotel Castel Dracula. One of my fondest memories was helping a local farmer dig his horse-drawn cart out of a snow drift along the highway.
Anyway, Romania is absolutely worthwhile to visit, especially if you have already visited the Western Europe