First, you must file your tax return as married if you are legally married on December 31 of the tax year, regardless of the status of your spouse.
Second, your spouse must file a US tax return if she is considered a US resident for tax purposes. She is a US resident for tax purposes if she has a green card and lives anywhere in the world, or if she lived in the United States for at least 183 days of the year. This is called the substantial presence test and there are a couple of other items, but the main one is living in the United States at least 183 days of the year.
Or, you can choose to treat your wife as a US resident for tax purposes even if she did not live in the US. This allows you to file as "married filing jointly", which has lower tax rates and better deductions than "married filing separately." But if you treat your spouse as a US resident for tax purposes, you must report and pay US tax on all her world-wide income, including income in her home country. She will get a deduction or credit if she paid foreign tax on the same money.
Third, if your spouse is eligible for a social security number, apply for it and don't do anything until she receives it.
So for 2018, your spouse was a non-resident. You must amend your 2018 return to change to married filing separately or married filing jointly. Remember that if you file jointly, you will get better tax rates in the US but must list her world-wide income, so only you can test which way works for you. If you amend to married filing separately, give your spouse's name and use a fake SSN to get the program to print your return. Erase or white out the fake SSN and write "NRA" (non-resident alien) in the space for her tax number. Mail the amended return. Don't forget to also amend your state return.
If you amend to married filing jointly, your wife needs an ITIN, international tax ID number, or a SSN. If you have applied for an SSN, wait for it to arrive before you amend to file a joint return. If she is not eligible, you will print the joint return and also download and print a form W-7, ITIN application. Take the tax return, the W-7, and any proof required in the W-7 instructions, to an IRS service center, or mail them to the address listed in the W-7 instructions. After the IRS issues the ITIN, they will process the amended tax return.
For 2019, you will file another amended return to change from single to either married filing separately or married filing jointly. If separately, use the same "NRA" procedure. If jointly, use the same procedure I outlined above, except that you won't re-apply for an ITIN, you will wait until the IRS issues the ITIN from the 2018 amended return and use it on the 2019 return. (Or, you will wait for the SSN and file both amended returns when it arrives.)
