Brasov is a beautiful city located in the foothills of the Carpathian Mountains and is roughly two hours by car from Odorheiu. We highly recommend the trip if you can get away for an evening or want to head there for lunch.
Sibiu is a gorgeous Transylvanian city worth a day trip or overnight stay.
Turda Gorge contains one of Romania's richest and most scenic landscapes. More than 1,000 plant and animal species (some rare or endangered, like the wild garlic or some eagle) live there.
Top Gear named Transfăgărășan the greatest driving road in the world. The views from the top of the Carpathian Mountains are breathtaking.