1. create an account on cloudflare
2. On the right sidebar click
   Domain registration
   then
   Register domains
   
3. search for a new domain name and click the
   Purchase button
4. if it asks for a plan, I pick the
   Free plan
   at the bottom
   
5. fill out the registration information and click the purchase button at the bottom
   
   

You have a domain name, congrats!


But you may want to do the following too:

• I do not click on the
  Prevent bots
  button as I suspect it adds an annoying cloudflare "are you a human" checkbox to your site. Not sure and you might want it.
  
• click on the
  configuration tab
• enable
  DNSSEC
  
• on the right sidebar click
  Domains
  , then find your new domain from the list and click the three dot menu and select
  Configure DNS
  
• click on the link
  "set up restrictive SPF..."
  to prevent email spoofing
  
• on the right sidebar click on
  manage domains
  and
  your new domain
• click on the
  Submit
  button to add the records
• on the right, click on the
  Back to Domains

You can view and manage your DNS records by...

• on the right sidebar click on
  Domains
• click on
  your new domain
  
  this overview page has several links to explore
• to see your DNS recods click on the link that says
  Go to DNS

That should be a good start. I have a page on how to create a website that can use this new domain name.