Time of year?
Romania is fascinating, as are the other central/eastern-European countries you haven't yet visited--Poland being surprisingly easy for independent travelers to manage. I wouldn't necessarily suggest trying to squeeze Romania and Poland into a one-month period because I think it would be rushed; I spent 5 weeks just in Poland last year.
Costs in Romania were extremely low at the time of my 2015 visit, and I sort of doubt that they've increased hugely since then. There are at least two rural areas of Romania that are not practical via public transportation and probably challenging by rental car from a navigational perspective: Maramures in the northwest (my favorite) and Bucovina in the northeast. I would recommend hiring a car and driver to spend at least two full days in Maramures (three would probably be totally worthwhile) and at least one full day in Bucovina (two would allow for more exploration). I think five days or so spent in those two areas, staying in rural B&B-style accommodations, would be totally affordable for a group of three, and I'm confident you'd find it fascinating.
During my visit to Romania I took a 2-day trip to Maramures that was organized by the Retro Hostel in Cluj-Napoca. There were three of us, as I recall, plus the driver in his nice air-conditioned car. The driver worked as a history teacher during the academic year, and I felt he gave us good historical background on what we were seeing. I am sure the Hostel would be glad to arrange a private tour for you and your daughters, and the "Retro Tours" section of its website might be helpful to you in identifying other places you'd like to see either on your own or as part of a private tour.
The places I had no trouble visiting in Romania via public transportation were Bucharest (not the nicest capital city you have seen, that's for sure; Ceausescu tore down most of the historic buildings), Cluj-Napoca, Brasov, Sibiu, Timisoara, and Sighisoara. I used a taxi to and from Peles Castle, but I think that may have been just for reasons of schedule. There's limited rail service in Romania and it tends to be slow. For planning purposes, I'd suggest estimating you will cover only about 30 mph via train or bus.