What is the correct order for obtaining a Tourist visa, Tourism extension, and Extension for visiting Thai family in Thailand?

As a rule that's how it's been, you have to take any "regular" extensions on the stamp you're here on before you can apply for an extension for a different reason

BUT

You should check with your immigration office,

a few started saying once you took the 30 day extension you couldn't get the 60 day visit thai family one (which is just a made up rule and not in writing anywhere)

AND

a few other ones started saying you can only get the 60 day visit family extension from a Non-Imm type visa, not a free stamp (visa exempt) or tourist visa entry

That's why I said CHECK with your office well before you go for the 30 day extension to see what's what where you're applying