Romania

Romania is one of those places where using a private driver might be warranted. And they're affordable. The backroads of this country appear to be incredible.
Google: "Romania private driver" and you can read all about the experience.
One is; http://covinnus.com/

There are lots of itinerary ideas here (including mileages):
http://romaniatourism.com/sample-itineraries.html

I don't think you necessarily need a driver or being part of a tour, although either option is remarkably inexpensive. Some folks on this forum have reported driving there and not having any issues. Many (if not most) trains are slow (30 mph or so) but the country is very well-connected via rail, in case you opt for public transport between some sites where it makes more sense.

Romania isn't exactly "Beyond Europe".

I used a combination of trains and public buses when I visited Romania for 17 days in 2015. It's a fascinating country. I don't think it's practical to visit Maramures in the NW or Bukovina in the NE via public transportation. Another issue to be aware of is the odd practice of requiring a telephone reservation for some inter-city buses. You sometimes can't just go to the bus station and buy your ticket a day or two ahead of time to guarantee your place; you have to make that phone call, or get someone to make it for you. If you skip that step, you may find yourself turned away when you try to board the bus. Otherwise, the buses are fine.

I stayed in Timisoara (cradle of the revolution, pretty historic district, infrastructure more westernized than in some other cities), Sibiu (another nice historic district), Cluj-Napoca (ditto, and a good base for hooking up with tours of the wooden churches and other folkloric aspects of Maramures), Suceava (if it had a pretty historic area I didn't find it; I was there to pick up a one-day bus tour to the painted monasteries in Bukovina), and Bucharest (scattered historic architecture and I think museums may be of interest). I recommend all of those stops (Suceava only as a base for Bukovina), but don't plan too much time in Bucharest. You have to walk past a lot of dull stuff to find scattered attractive architecture, since Ceausescu loved to knock down anything with character.

I took day trips to Shighisoara (relatively pure medieval district but not undiscovered), Oradea (some nice art nouveau architecture, not as well restored as some other cities), and Brasov (really nice historic area). All worthwhile. I felt Iasi was skippable.

The Retro Hostel in Cluj-Napoca has a large number of tours on its website. That's how I hooked up with the really good 2-day tour of Maramures I took. I wasn't staying at the hostel, but that did not seem to matter. The guide on that particular trip was Florin Merciu.

I agree with David. We have used the owner of this site a couple of times http://covinnus.com/

One of my favorite guides over the years.