Tax Year Prior to 2020: Why will it not accept my return says it will be 2/3/2020 try back then

It depends on the state you live in and when that state will begin accepting/processing returns.

I am getting the same message("2/3/2020. try back then.") when I try to e-file using TurboTax. For example, Georgia will not begin processing state tax returns until 2/3/2020.

If you're not in Georgia, check with your state's website and see when they will begin processing returns.

Okay - mine also said 2/3/2020 Question is will I be notified when the e filed return has been accepted? Anything else I need to do meanwhile?

It all depends on what state returns you are trying to file as well. Even though the IRS has opened and started accepting e-filed returns, some states have not. You can check the status of when you state forms will be ready for e-filing by clicking here: https://ttlc.intuit.com/community/forms/help/state-forms-availability-table-for-turbotax-individual-personal-tax-products/00/26227. Once they show available, that means they can be e-filed.