I'm past the newbie stage, but I remember that it was difficult for me to add a new domain to existing hosting. Even courses or advice with step-by-step instructions that show you how to start with a new domain and new hosting, don't show you how to add a domain to existing hosting.
Since Hostgator (and other web hosts) offer a plan with unlimited hosting (the Baby plan), it's a pity to pay extra for it (hosting) when you can have it for free.
Anyways, here's how to add a new domain to existing Hostgator hosting. While these instructions are for Hostgator, I think that it's probably about the same way for other hosting companies.
After getting the domain name (say, Namecheap or GoDaddy), set the nameservers to point to Hostgator. (You probably know how to do that, since this is not your first domain, but here's some quick instructions:
If you'll check your e-mail (at the e-mail address that you gave to Hostgator) you'll probably find an e-mail from Hostgator. In the e-mail, you'll find some information about nameservers.
Nameservers look something like this: ns1005.hostgator.com and 1006.hostgator.com.
Now, go to your domain registrar and sign in with your account information (in this case Namecheap. It's probably pretty similar with other domain registrars.)
Click on "My Account", then on "Manage Domains"
On the left hand side, click on "Your Domains" and doubleclick on your domain name. Again, on the left hand side click on "Transfer DNS to Webhost", then on "Specify Custom DNS Servers".
Go back to your Hostgator e-mail and copy & paste the nameservers one at a time in the boxes provided. Click on "Save Changes" and you're done with pointing your Nameservers.
In your Hostgator cPanel, (an easy way to get to your Hostgator cPanel is from your Hostgator e-mail. There's a link there.)
Scroll down to the box labeled "Domains" and select "Add-on Domains".
You put in your new domain name. Remember not to put "http://"
or "www", just the name and .com (or .info, etc.). The SubDomain and Document Root boxes will be filled in automatically. Enter a password twice, then press "Add Domain". You're done!
I hope this helps you!