You could take a Cab, hire a private taxi or drive. (If you are an international tourist you may want to check the requirements for Driver’s licence and road regulations with the Maharashtra State Road Transport department.)
You could take a Maharashtra State Transport Bus.
You could take a Bus operated by Private operators.
You could take a Central Railway train of the Indian Railways from Mumbai (Bombay) V.T. station to Manmad station and then take a bus to Shirdi. State transport buses are available from Manmad to Shirdi.
For Fares, Schedules and other details contact your local travel agent, the Government of India Tourism Office, the Maharashtra State Tourism Board Office, the Central Railway booking office or the Maharashtra State Transport booking office.

You could first go to Mumbai by train, bus or an airplane and then follow the steps listed above.
You could take a train from the city of origin to Manmad. Not all trains go directly from point to point. You have to check if the train goes directly to Manmad or if you have to break your journey and change over trains. From Manmad you have to take a bus or cab to Shirdi.
You could drive, hire a cab or private taxi to take you directly to Shirdi. (If you are an international tourist you may want to check the requirements for Driver’s licence and road regulations with the Maharashtra State Road Transport department.)
For Fares, Schedules and other details contact your local travel agent, the Government of India Tourism Office, the Maharashtra State Tourism Board Office, the Central Railway booking office or the Maharashtra State Transport booking office.