I am having problems trying to figure out how to register your own dns. I know how to register a domain name and I have registered one. Lets call it first.com. The domain name I registered I want to referrence as my actual DNS then I want to register more domain names and have my DNS as their source of authority but can't figure out how to do that.

for example

I want to register second.com and third.com and point their dns resolutions to ns.first.com so that if I have to change the ip of second.com or third.com without having to change something on my domain register company.

Can anyone help me?

If you need clarification let me know.

the dns registry is related to domain name registeration. I don't think if you can do the one without the other.
If you have a first.com alredy registerd, then it's dns registeration is being hosted by your registerar/web hosting company.
Domain name regiesteration requires that at least 2 independent dns servers are able to respond to queries for your name, so your first.com enteries will have to be cloned by another server.
So the owner of those dns servers will have to put your name-ip-address binding into their server. Earlier universities used to do that gratis, but these days you will have to go to a web hosting company (.. there might be others .. even verisign etc. might do it). You most likely will have to pay for it.

Your DNS servers can really be anywhere, I never heard of a rule where you are required to have 2 - I've ran only 1 for years now. This is just my example, I registered with tucows, and https://manage.opensrs.net/ is where I go to set my domain name server. You enter a username and password you received when you register your domain and you get a setup screen to enter the ip address of 1 or more name servers. I'd assume most others work similar. You'll have to call your registrar and get the information if you don't already have it.