I think it would be best for me to file an extension, but how can I know what I will owe (so that I can pay on time) if I haven't yet filed? I'm confused!

You should make a best estimate of what you think you may owe.  Possible ways to estimate include:

• Complete as much of your return(s) through TurboTax and base your estimate from those results.
• What did you pay last year in federal and state taxes (this may be a good gauge if your income situation hasn't changed much from last year).
• You can use the IRS tax tables (find your income level and filing status) to see what is typically paid at that level: https://www.irs.gov/pub/irs-pdf/i1040tt.pdf
• If you overestimate & pay extra tax, the IRS will refund any over-payment.
  

You'll be able to file an extension thru TurboTax by following these steps:

1. Click on the Federal Taxes tab
2. Click on the Other Tax Situations subtab
3. Scroll down to Other Tax Forms 
4. Select File An Extension

The important thing to remember about extensions is that they only extend your time to file the return.  If you owe money, then the IRS and states expect you to pay by the normal tax deadline -- there's no extension for paying. So if you're filing an extension and you believe you'll owe, then you'll want to mail a check to the IRS or state.  

On the other hand, If you normally receive a refund, and you don't anticipate any change, then you don't need to send payment.

In regard to your state return: Some states will piggyback off the federal extension, so there's no additional state form to file.  Other states require you to submit their own extension forms.  TurboTax will walk you thru the state extension as well as the federal process to ensure you follow the requirements of your state.

If you're curious about your particular state, here's a list of extension requirements by state: https://www.irs.com/articles/state-tax-extension-information