In Cluj we stayed at Piccola Italia, a very reasonably-priced hotel that we loved. The staff is extremely helpful with luggage, information, etc. It's located about 10-15 minute walk from city center, in a quiet residential neighborhood.
In the Apuseni area just west of Cluj are many villages where homestays are available. We stayed a full week in Sancraiu, which has many homestays. It's a small village, where your main activities would be experiencing local life and hiking around. These villages lack restaurants, and other amenities, so you either buy food at a store, or are provided by meals with your hosts. The Apuseni area is about 1 hour west of Cluj, the largest town in the area is Huedin. There is a train to Huedin, and you'd need to arrange transportation to the village you intend to stay if you don't rent a car.
One source of information is: www.greenmountainholidays.ro.