Nothing is working for me past creating my account.

There is no appwrite-provided subdomain. I can't create my own.

I stopped there and came here.

I am not going to write a function that I can't access (without some sort of subdomain)

I'm confused. You plan to use appwrite with an SDK?

Today my goal was simply to do "hello world" with the platform to get a sense of how it works.

You need to create the function in order to assign it a subdomain and in order for it to work.
For example, this is a hello world one I deployed right now from the existing template: https://687123c000313108f804.fra.appwrite.run

In order to try, you can choose the existing hello world template

Can I use my own subdomain?

Sure

And it's free of course

But you need your own domain for that

I clicked "functions" and then "create function" and then "browse all templates".

In the search template box I typed "hello". No results found.

Choose one, for example, the node starter function that is pretty much a hello world one

Does appwrite need to manage DNS for my apex domain if I'm going to add a subdomain?

The only behavior I get from your platform when I choose "add domain" is for an apex domain -- even if the domain name I give it has 3 or more components.