How do I know if I've filed my federal and state taxes already? I'm not sure if it went through when I completed them through the program.

You can check to see whether your tax return was filed by simply re-opening your return or signing back in to TurboTax. Your filing status should be the first thing you see, it will either say e-file or Ready to mail for your status. If you do not see your filing status, then you will need to revisit your return and complete the filing process.