I have started private therapy practice and get a 1099NEC. Do I need to pay quarterly fed tax & self-emp taxes (medicate+social sec) at irs.gov/payments/direct-pay ?

Pay quarterly. You will enter those payments into TurboTax when you do your 2023 return. 

The tax system is pay-as-you-go. With an employee, wther are amounts withheld from each paycheck. A self-employed individual has to pay estimated taxes. Your SE taxes are 15.3% of the profit. Your Federal tax is according to your bracket.

If you wait until the end of the year, you will probably face an underpayment penalty. You are expected to make four quarterly payment and keep track of those payments yourself. You will pay with Form 1040ES and enter the amounts paid on your tax return the following year. 

Yes, you need to make quarterly estimated payment of your taxes, both federal and state. When you do your taxes next year Turbotax will ask you in the Deductions and Credits section if you made estimated taxes during the year.

Also, do I really need to complete the 1040ES form? Or is that just to estimate what the quarterly payments need to be?

Thank you!  I don't see that filling out the 1040ES form is required when I review the payment process at https://directpay.irs.gov/directpay.

Also, it looks like I need to pay both fed and self-emp tax quarterly.  Is that right?  So, that would be about 30% of my income.

Filing 1040-ES is not REQUIRED, but if the IRS determines you significantly underpaid your taxes during the year, it could level a penalty in addition to your taxes. The 1040-ES helps you avoid this penalty.

Yes you should file 1040-ES for both federal and state taxes.