I had a balance due for my federal and state 2017 taxes. I paid both balances in 2018. Can I claim these federal and state taxes on my 2018 return?

Unfortunately, you cannot deduct the federal taxes you paid. However, you can deduct state taxes as an itemized deduction on Schedule A. If you choose to itemize your deduction to claim state taxes you will not be able to take the standard deduction. You should calculate both the standard deduction method and the itemized method to determine which is most beneficial to you.