how does the actual process of registering a domain work?

It's actually remarkably simple.

You send in the request to the registrar who adds the domain name and the associated nameservers information up to the root nameservers and keep a copy of the information for themselves.

The nameservers that you give in the domain registration process are not actually FQ server names - they are just place holders. Theoretically, there is no reason you couldn't name your nameservers "joe", "fred", "mike" or whatever.

Regardless, the root name servers are told that the nameservers you register with are the authoritative nameservers for the domain you just registered. That's really all there is to it.

Your next questions is probably, "well, how does the dns info resolve then?"

When a dns server makes a request for a domain, if it doesn't have the information cached locally it passes the request up the line to see if the root nameservers know where to get authoritative information from. If not, the whois database is queried to find out where the authoritative details about the domain can be located.

This is a highly simplified version of what actually happens, but it's pretty good for the purposes of illustration.

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so instead of paying 10 bucks a year .. cant you just have your own dns ?

if you setup redhat linux 8.0 it has an option of being a dns
is this all you need? or do you have to tell some "master dns" that you are a dns