From the standpoint of superficial visual interest (I like pretty buildings), I'd have a tough time choosing between Timisoara and Cluj-Napoca. I highly recommend both, but there are differences.
Though both are comparatively westernized, Timisoara is perhaps more so; the first time I ate quinoa, it was in a casual restaurant in Timisoara. Timisoara is where the revolution began. There's a very interesting old-fashioned museum on the subject. The building is crumbling, there are no high-tech displays, but the explanatory material is mostly (perhaps totally) translated into English and is very informative. The museum requests a donation. Please be generous.
The lovely Cluj-Napoca is also important as the jumping-off point for the very interesting area of Maramures with its wooden churches, carved gateposts, Merry Cemetery and political prison (English handout available to borrow). It's a shame to miss Maramures, but it needs a couple of days and one night. The Retro Hostel in Cluj-Napoca runs tours, but they operate on demand, so you'd have to be lucky in your timing. The price was more than fair in 2015: about US $100 plus $20-25 to the B&B where we spent the night and had three (!) quite good meals of mostly home-grown/raised food. You do not have to stay in the hostel to sign up for its tours.
Peles Castle is interesting, though not old. The towns of Sibiu (another pretty one) and Sighisoara (hilltop and medieval-looking; more intensely touristy) are also worthwhile. I liked Brasov, too, but had only about 90 minutes there and don't know how it stacks up from an historical-site standpoint. Really, Romania has a lot of fascinating cities and towns. You're right to choose just a few so you have time to appreciate the ones you visit.
I think the time you'll want in Bucharest will be driven to a considerable degree by what museums you want to see. Before my trip I expected the city not to be very attractive because Ceausescu had obliterated most of the historic architecture, so I planned just an overnight stop. I did find quite a lot of attractive buildings, though most were 19th/early 20th century. But I had to work hard to see them, because they were rather widely scattered. That's how I spent my time, pounding the sidewalks of Bucharest, rather than visiting museums, so I cannot provide any tips on that score. I'd suggest taking a look at TripAdvisor's Things To Do list as well as a guidebook to Romania to figure out your level of interest in Bucharest's museums
If you had posted an open-ended question, "How should I spend 9 nights in Romania?", I'm not sure how I would have responded, but two days for Maramures would definitely have been included, which would also have required Cluj-Napoca. For the rest, I think Timisoara, Sibiu and Brasov are of fairly equal worth as tourist targets unless you have a powerful interest in the Communist era and the revolution (as I do), in which case I'd go with Timisoara. Otherwise, I'd probably choose Sibiu or Brasov since either of those allows a side-trip to the other one, Peles Castle or Sighisoara.
Without the Maramures tour, I'd do Timisoara plus either Brasov or Sibiu.
Between Sibiu and Brasov I'd be guided by the train/bus links available; you won't make rapid progress on either--perhaps 30 mph, so if you haven't already looked up the schedules you should do that now. Because of transportation links to the outside world, Bucharest is pretty unavoidable.
Perhaps:
July 1: Fly Stockholm to Bucharest (2 nights, or 3 for more museums)
July 3 or 4: Train to Brasov or Sibiu (3 nights, allowing for a possible side-trip)
July 6 or 7: Train to Cluj-Napoca (3 or 4 nights, with 2-day Maramures* trip)
July 10: Fly home
- Substituting Timisoara if Maramures is a no-go.
I'm almost out of room and will put some food comments in a separate post.
