What documents are required to extend a tourist visa in Thailand, and can I switch to a family visa afterward?

specifically for those in Bangkok

You may be able to get the Family Visit 60 days extension of stay from your local Thai Immigration office. Usually approved for one Entry into Thailand. Visit your local Thai Immigration office for their requirements for the Visiting Thai Family extension.

At most offices you can go to get a 30 day extension on a 60 day free stamp (visa exempt) or 60 day tourist visa entry when you have 30 days or less left on your current stamp. Some offices (Jomtien/Samui, etc) only allow you to get the extension when you have 7 days or less.

Are you here on a 60 day free stamp entry OR did you buy a tourist visa online from the thai consulate in your country before you came here?

I ask because IF you're in Bangkok you go to different offices depending on what you stamped in on. People with real tourist visas go to the main office at the government complex (Building B) on Chaengwattana Road, and people with visa exempt entries go to IT Square Laksi Plaza (3rd floor).

As far as can you switch to 'family visa' that depends on whether you meet the requirements and if your immigration office can issue in country 90 day Non-Immigrant Type O visas or not.

This is a general list of what can be required at immigration offices for a 30 day extension on visa exempt or tourist visa entries.