If filling out name in w9 for the first time. Should they put (1) just first and last name, or (2) first , middle initial, last name, or (3) full name spelled?

It doesn't matter. They should put their name on the W-9 the way they want the payer to put it on checks and 1099 forms.

You are talking about Form W-9 that they use to provide their tax ID number (SSN or EIN) to a client or someone else (but not an employer) who is going to make payments to them, is that right? Who are "they," and what is their connection to you?

Thanks for your reply!

"They" is my child.

READ the W-9 instructions ... there is one for each line :   https://www.irs.gov/pub/irs-pdf/fw9.pdf

While it's not a requirement in writing anywhere I'm aware of, if you enter the individual's name just as it appears on their social security card, you'll be fine.