Romania and Hungary

I suggest you spend more time in Transylvania and less time in Bucharest.

Bucharest old town and pedestrian district is fun and lively. Authentic Romanian meals in fantastic atmospheres are available at Caru' Cu Bere and Hanul Lui Manuc: http://www.carucubere.ro/, http://www.hanumanucrestaurant.ro/ro/

The Palace of Parlaiment is the don't miss attraction in Bucharest. I also recommend the National Village Museum near King Michael Park and the Arcul de Triumf. http://muzeul-satului.ro/

But, there's not enough in Bucharest to hold your attention for an entire week.

Consider also a visit to the Danube Delta.

Peles Castle in Sinaia is a must-see, en route to Brasov from Bucharest. Closed on Mondays, I believe. Easy access from the main train line between Buc and Brasov. http://peles.ro/

In Brasov, I love the Hotel Kolping. It's near the old town and the main Piata Sfatului, and literally right next to the mountain. Easy access to the main paths that run along the old city walls and bastions and the cable car that takes you to the top of Mount Tampa. From the back gate, you have immediate access to back trails that explore the gorgeous and less-visited parts of Mount Tampa. http://www.hotel.kolping.ro/

I would suggest a night or two in Sighisoara, the long way between Brasov and Sibiu. The UNESCO world-heritage-site fortified churches between Brasov and Sighisoara are much easier to visit if you have a car. The best fortified churches are in Viscri and Saschiz. There is also a fortified church in Prejmer, with easy public transport just outside of Brasov. Another fortified church in Rupea is right on the main road/trainline between Brasov and Sighisoara. Gelu Trandafir is a local auto-tour guide who offers personalized tours to the areas you are considering. http://proiect.tripdor.ro/ Consider to do the Transfagarasan Hwy tour, into the Fagaras mountains. https://en.wikipedia.org/wiki/Transf%C4%83g%C4%83r%C4%83%C8%99an

From Sibiu, visit Corvin Castle (in Hunedoara), one of my favorite castles in Europe. http://www.castelulcorvinilor.ro/
Also, the Citadel at Alba Iulia is an easy day trip and consider to visit the Astra open-air museum just outside of Sibiu. http://muzeulastra.ro/

Also, consider to visit the Zalan Valley and Miclosoara. Prince Charles (from UK) and local Hungarian Count Kalnoky have created an authentic, all-inclusive tour that includes a deep-dive into the local culture and countryside: https://zalan.transylvaniancastle.com/
https://www.youtube.com/watch?v=7PQJ_kSLt08
Zalan Valley is really remote. You can arrange transfers from Brasov (closest big city), Sfantu Gheorghe (closest town with a decent station), Viscri, Sighisoara, Bucharest, and many other locations.

I recommend you watch the Wild Carpathia series to get some more ideas and background for your visit:
http://www.wildcarpathia.tv/

Timisoara is an easy stop-over en route from Sibiu/Romania to Budapest. Lots of '89 Revolution and Hapsburg/Austro-Hungarian history.

I loved the two-day tour I took to Maramures in 2015. It was organized by the Retro Hostel in Cluj-Napoca, but It would be affordable to hire a private guide to visit the area, which just can't be seen by public transportation. It's a very distinctive area where folk traditions live on. There are wooden churches, wooden gate posts in front of some houses, the Merry Cemetery and more.

Trains in Romania move at about the same speed as buses. I figured I covered about 30 mph. You may need to take some buses, because, the rail network is rather sparse.

The procedures may have changed since 2015, but two of the bus trips I took required a reservation made by telephone. It wasn't possible simply to go to a bus station and buy a ticket in advance. Two kind Romanians made those calls for me. I'm sure I needn't tell you that you should not expect bus-company employees to speak English unless they are quite young.

Based on my experience, you may want to tilt toward touristy restaurants in Romania. The general population doesn't have the money to eat out. However, the large dough rings sold everywhere (with various fillings and coatings) were wonderful. Often right out of the oven.

Congrats on the upcoming trip! I'm sure you'll have an amazing time!

Just throwing out some random things...

Certainly check out Peles castle - it has to be one of the most visually stunning castles around (not an imposing fortress type castle, more of a fairy tale Neuschwanstein-type place) from both the outside and in, and set in a really gorgeous area. I'd imagine you plan to go there anyways, as it's not far from Brasov. I do believe the interior of Peles closes for a part of winter, so keep that in mind depending on when you're taking your trip. While I only spent a day in Brasov myself, I do wish I'd spent more time there. It does seem like a great place for lazy days and lots of strolling around! On the subject of imposing fortress type castles, though, Rasnov has one and is also a pretty neat little town - like Brasov, just really picturesque areas, charming streets, fascinating things to look at. And heck, even Bran castle is neat. Sure, there can be crowds and the whole not-really-tied-to-Dracula thing is kinda funny, but the castle itself was an incredibly fun visit for a castle nerd like me - you get to walk all around inside it and it's a very up close and personal look at what life was like for its residents. So its real merits have nothing at all to do with Dracula.

Not sure if you guys are wine drinkers, but Transylvania has lots of options on that front for wineries, wine tastings.

In Bucharest itself, Ceausescu's Spring Palace is worth a look. Its interior is just so over the top, elaborate, and ornate, and the (English language) tour was very well done and informative - small group, guide takes you around and through the house and gives a lot of information and context about what you're seeing. I also really liked Mogosoaia Palace in Bucharest - the buildings and grounds are gorgeous.

When it comes to the Palace of Parliament, I wish i hadn't committed so much of a day to touring it. On the one hand I suspect I might have regretted not seeing it had I gone to Bucharest and skipped out on such a landmark, but on the other it was quite a lot of time waiting in lines for a quick little stroll through it.

I think Bucharest is very well suited to your sights + lazy days approach. I did the same when there, and found the city to be extremely walkable with fascinating stuff to see and great cafes to hang out in all over. Say, south of Kisseleff park and stretching to and around Old Town and the university. Old Town is packed full of cafes and restaurants, and if you stroll south from the old Communist Party headquarters along Calea Victoriei there are neat places to stop and hang out. Having said that, I wish I'd given more time to Transylvania and various towns, cities, and sights there.

In terms of getting from Bucharest to Brasov, I used a private guide. They're very affordable there, and I liked the flexibility of being able to occasionally stop and check things out. Mind you, the highway from Bucharest to Brasov can get quite busy so if you were to travel that way you'd want to avoid competing with all the weekenders from Bucharest making their way to Brasov and the surrounding areas.

Yes to Castelul Peles. If you are a photographer, go ahead and spend the extra money for the Photographic permit. I did not and regretted it.

In Bucharest, we went on the "free tour". Very interesting and informative. We stayed in the Umbrella Hostel - inexpensive, not far from downtown, with a private room. It was close to a restaurant La Mama, which we enjoyed. We also went to Caru cu beri (beer wagon). Very fun, touristic. Some of the staff do dances.

We took the train from Timosoara to Budapest. It was a great way to go. We loved Sibu, though going off peak season was a plus.

My 13 year old son and I had a great walking/driving trip around the major communist sights within Bucharest with this company:

http://unveilromania.com/romania-tours/bucharest-tours-communism/

It was a one on one guide and Emma, our guide, was fantastic. She was born and raised in Bucharest and knows her history incredibly well, her English was impeccable and because it wasn't a group tour there was much more fluidity around where we could go and what to see and do.