From Cluj-Napoca, take a train to Turda and then a local bus to the salt mine. The journey is approximately 1.5 hours. You can buy online ticket here on the official website. The entrance fee to Turda salt mine is around 10$ USD per person.
Turda Gorge
From Turda, take a local bus to Hășdate village, then a short walk leads to the gorge entrance.