You will easily get bus from Somnath to Dwarka
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You will get auto for shivrajpur beach
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You can take a bus from Somnath to Dawrka that you can book online also. Then Dwarka to Okha would be some 20kms. Then at Okha jetty area (if you're considering to visit Beyt Dwarka too) you'll find some autos and cars to Shivrajpur beach. All from Dwarka to Shivrajpur will take around 1.30 -2 hours.
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Yes it's Your Choice. You can Travel By Train Somnaath To Dwarka. you can check the time on website and You Can Choose Bus Or car on road trip there is two way available. one is by porbandar and second one is by rajkot. you can check both root in map and select your best. For Visit Nageshwar You Can Take Auto Rikhsa or car from Dwarka to visit Nageshwar. For Shivrajpur Beach You have to take Auto Rikhsa or car from Dwarka to Reach Shivrajpur Beach.