Should I file prior year's taxes first? Or should I file this current year's taxes first?

If you do not have any credits that will carry over from year to year, you can prepare them in any order.  But you can only use online for 2022.  Earlier years must be prepared using the downloaded software and the returns must be mailed.

Each tax year has to be filed separately using the forms for the specific tax year.  They cannot be combined in any way--do not even put them in the same envelope when you mail them.  Software for past years is available back to 2019.  Any earlier years can only be prepared on paper forms by hand--and those refunds have been forfeited now.

The current online program can only be used for a 2022 return--no other year.  And if you did not file a 2021 return you will have trouble e-filing 2022.  When it asks for your 2021 AGI try using zero.  If zero does not work then you have to print, sign and mail 2022

Online preparation and e-filing for 2019 and 2020 and 2021 is permanently closed.

To file a return for a prior tax year

 If you need to prepare a return for 2019, 2020 or 2021 you can purchase and download desktop software to do it, then print, sign,  and mail the return(s)

https://turbotax.intuit.com/personal-taxes/past-years-products/

You may also want to explore purchasing the software from various retailers such as Amazon, Costco, Best Buy, Walmart, Sam’s, etc.

Remember to prepare your state return as well—if you live in a state that has a state income tax.

https://ttlc.intuit.com/community/state-taxes/help/how-do-i-contact-my-state-department-of-revenue/00/25573 

When you mail a tax return, you need to attach any documents showing tax withheld, such as your W-2’s or any 1099’s.  Use a mailing service that will track it, such as UPS or certified mail so you will know the IRS/state received the return.

Federal and state returns must be in separate envelopes and they are mailed to different addresses.  Read the mailing instructions that print with your tax return carefully so you mail them to the right addresses.

Note:  The desktop software you need to prepare the prior year return must be installed/downloaded to a full PC or Mac.  It cannot be used on a mobile device.

If you are getting a refund, there is not a penalty for filing past the deadline.  If you owe taxes, the interest/penalties will be calculated by the IRS based on how much you owe and when they receive your return and payment. The IRS will bill you for this; it will not be calculated by TurboTax.

If you can file a correct return for 2022, you should go ahead and file it. Then go back and prepare and file the prior year returns. If necessary, you can amend the 2022 for any corrections. You will be able to e-file an amended 2022 on Turbo Tax software as well.