If you have some time, you can watch this video. Take the comments with a grain of salt but it does highlight most of Romania’s marvelous places you can visit. You won’t probably have time for all of these but the following sites are located near Cluj-Napoca:
Turda salt mine: one of the most impressive salt mines in Europe)
Turda gorge (and Tureni gorge): impressive gorges and ladscapes for those that like hiking (easy routes through the gorges, Turda gorge easier with higher walls than Tureni gorge but Tureni gorge is wilder in nature in the gorge)
Rimetea village: an impressive village at the base of a mountain cliff
Sighisoara: inhabitted middle age city fortress; one of the most visited cities in Romania
Lake Tarnitza: a beautiful lake just about 20 km from Cluj; a full day is good to have to relax in those amazing landscapes
Banffy Castle: a very interesting place hosting one of the famous music festivals Electric Castle; there is also a smaller castle along the way, the Banffy Castle in the village of Rascruci which is newly renovated
Apuseni Mountains: marvelous mountain chain with a lot of hiking routes with many caves and beautiful landscapes; probably at least one full day needed for a hiking route
Alba Iulia, Sibiu, Brasov, Timisoara: cities worth a visit ordered by the distance to Cluj-Napoca
Now, these are attractions relatively close to Cluj-Napoca. For sure if you want to adventure further away there are many beautiful places around Brasov, Bucharest is also worth a visit. Many of the attractions listed above you may be able to reach by public transport. However, renting a car or hiring a local transport company will get you faster to these destinations.
If you are spending more time in and around Cluj-Napoca, here is a good start for searching for attractions that you may wish to visit.