The easiest way is to take Motorway (M2) to reach there. If you are coming from Lahore side or Islamabad side, exit from Lilla Interchange and take the road that is leading to PinddadanKhan-Jhelum. Continue on that road for about 20 km and you will see a board and a road pointing towards your left which leads to Khewra Salt mines (roughly 6-7 km from that left turn).
If you don't have your own transport then you can reach from Jhelum side on public transport. Any bus from Jhelum or some other area that goes to Chakwal via Choa Saidan Shah should drop you at Khewra. May be someone else can suggest better in terms of public transport. Although I would advice you to hire a car with driver for a single day trip.
It would take 3 hours maximum to explore the mine on foot. Train ride is a unique experience too.