I moved to U.S. and received my green card in September 2018. Purchased health insurance through Covered California and received premium deduction. How should I file?

As a green card holder, you can file a 2018 full year tax return in TurboTax.  You will need to declare your worldwide income for calendar year 2018.  If you had foreign income in 2018 before you moved to the US, you will need to include that income, however you can use foreign tax credit to help offset any double taxation on foreign income.

You will need to enter an exemption for the months in 2018 that you did not have health insurance. 

Yes, the tax penalty for not having health insurance in 2018 can be waived if the individual was a U.S. citizen or resident but lived in a foreign country or U.S. territory for at least 330 days within a consecutive 12-month period that includes all or part of 2018.