I changed my last name this year and my W-2 form has my old last name. How should I file my taxes? With my old last name or new last name? Will this cause delays?

Use the name that matches your current Social Security card.  

The name on your tax return and the name on the W-2 do not have to match, only the Social Security numbers must be the same.

The name you enter on the tax return has to be entered exactly as is shown on your current Social Security card.

You need to file with the name that is on file with the Social Security office, regardless of what is on your W2's.