Recent US Green card holder not NRI, as came to US just 2 months back.
After coming to US, how can I Transfer money from my own savings bank account in India (No NRE or NRO A/C) to my recently opened checking account in US?
You are now an NRI (NonResident Indian) in the eyes of the Government of India and as such, not entitled to hold a regular savings account in India, even if you have been holding one all your life before you moved to the US. Indian banks are forbidden from letting NRIs hold and operate regular savings accounts but have the out that until the bank knows that you are an NRI, it doesn't need to do anything about the matter. So, if you never filed a change of address telling the state Bank of India (SBI) that you are now in the US (or sent them a change of address form telling them that your new address is your parents' home address in India, etc), SBI will let you operate the regular savings account without any hassles. What SBI will not let you do is transfer any of that money to your own account in the US.
So, as @Dheer says in a comment, the first thing you need to do is tell SBI that you are now an NRI and that you want to convert your regular savings account into an NRO account. They will "be happy to assist you" in doing so because they too are required by law or Reserve Bank of India regulations to change such accounts to NRO status as soon as possible. Of course, this being SBI, you will most likely have to go through incredible contortions and paperwork, sending them notarized copies of your green card which notarization might need apostille certification by the Secretary of State of the US State in which you reside, followed by certification from the local Indian Consulate that the Secretary of State certification is indeed legitimate (that is, it is not something that you printed out on your color laser printer) etc. Once you have done all this, you should follow the advice given by Dheer in his answer to Transfer Money from my account in India to my account in the USA