How do I get the dates I was employed from a previous employer?

The actual dates of employment can only obtained from the employer.

Your tax returns will not show that information.   Dates of employment are not on your W-2's---a W-2 only shows which YEAR you worked for an employer---and TurboTax only has the worksheets that show the income information you entered.  TT has no actual copies or images of your W-2's.  If you know the employer's name, then contact them for their record of when you worked there if you do not remember.   This is an example of records you simply need to keep for yourself.

other than your employer only paystubs might contain the info. Nowhere else.  

You will have to call and ask. 

You may need to get HR on the phone. 

You may need to request in writing if they won't do it over the phone. 

Less direct - Call as a new job and ask for dates of service and references.