First off, you need a domain or server; it depends.
The actual .com is a domain, so you'd need to buy one with a hosting provider. Then you'd need to read up on some tutorials for HTML, CSS or Javascript.
http://www.w3schools.com/html/html_intro.asp(That link has a lot of interesting info and examples)
Then you'd need to download a File Transfer Protocol client (FTP client) to be able to send the website over to the internet. It does take a lot of time, and effort, but there's a lot of easy to use website creators out there, such as Webs.
www.webs.comAdoptable pet sites need a database to store pet information and a lot of other features. MyAdopts is another example of an easy to use website creator, only it's more for adoptable pets.
http://www.myadopts.com/Feel free to PM for more info. :3