No reporting of India-sourced income or paying US income taxes on India-sourced income if you are considered a nonresident alien.
For any part of the year you are considered a nonresident alien (whether that is the full year or only part of the year), you will only include US-source income for that part of the year you are not considered a US citizen or resident alien. Therefore, you will not have to include your Indian wages (India-sourced income) or pay any US income taxes on any India-sourced wages unless earned during the part of the tax year you are considered a US citizen or resident alien.
Consult the US Tax Guide for Aliens to determine your residency status relating to filing US federal income taxes.
If you do not meet the two above tests or you have a visa exemption, you will be considered a nonresident aliens and will file Form 1040NR to report your US source income. Form 1040NR is not supported by TurboTax
TurboTax does not support dual status filings and they can be quite complicated.
For your First Year of Residency, the US Tax Guide for Aliens states that if you are a U.S. resident for the calendar year, but you were not a U.S. resident at any time during the preceding calendar year, you are a U.S. resident only for the part of the calendar year that begins on the residency starting date. You are a nonresident alien for the part of the year before that date.
Here is a link to the IRS website for Form 1040NR, U.S. Nonresident Alien Income Tax Return