As a Maine resident, Maine will tax all of your income regardless of where you earn it.  Therefore, when you enter your W-2 into TurboTax, you first want to treat all of your W-2 copies as if they were one:  enter the Maine amount as reported on the W-2 with a box 15 of ME, and then boxes 16 and 17 in their corresponding spots.  Next, you will click on "select another" and continue to add the state name from box 15 and the corresponding amounts is boxes 16 and 17.  If there are any local taxes withheld, then you will enter these amounts found in boxes 18 and 19, clicking "select another" for each different locality.  Box 20 is the locality ID, and then you add the state associated.  The screenshot below provides guidance.

What should happen is that the correct taxable amount will transfer to your Maine return.  However, you will want to prepare all nonresident returns first, that is the other states where you worked and will be taxed (MA and MO; NH does not have state income tax).  This is because on the Maine return, Maine will issue a tax credit for the amount of tax you must pay to each of these states for the amount of income earned in each.  This FAQ provides more information:  https://ttlc.intuit.com/replies/3302052

So… after adding Massachusetts in. Box 15,16,17 (from my mass. W2)in my Maine w2 … do I next enter all of the information for my Massachusetts w2 entirely including the box 15,16 and 17 as it states on the mass w2?

then when running my state tax returns I do the Massachusetts one first followed by my Maine income statement?

So… after adding Massachusetts in. Box 15,16,17 (from my mass. W2)in my Maine w2 … do I next enter all of the information for my Massachusetts w2 entirely including the box 15,16 and 17 as it states on the mass w2?

then when running my state tax returns I do the Massachusetts one first followed by my Maine income statement?

Yes, that is correct. Prepare your nonresident Massachusetts return prior to your resident Maine return. The credit you receive on your Maine return is for taxes actually paid to Massachusetts, not taxes withheld by Massachusetts. TurboTax needs to calculate the amount you actually pay to Massachusetts before determining how much the credit should be on your Maine return. 

If your W-2 lists income and taxes withheld for both states, TurboTax will automatically carry the correct state's information over to the state forms. You may need to tell it specifically not to include the other state's income. 

@Tatambe