Will I get the 2nd stimulus check and can I still get the first one if this past tax return was my first time filing on my own as not a dependent?

You may not get the second payment automatically if your status recently changed.  However, as long as you are not a dependent of another taxpayer in 2020, you should receive the full amount of both payments when you file your tax return.  It will be applied as a credit that will increase your refund or decrease your balance due. 

To claim the Recovery Rebate Credit in TurboTax, answer 0 for the two questions about the stimulus payments that will appear at the end of the TurboTax interview.  (Or select no, I did not receive any payments, if this option is available in your version.)

The questions are in the section Federal Review.  

Once the information is entered you will see your refund increase or balance due decrease, and TurboTax will enter the amount of the credit on Line 30 of Form 1040 for you.