Say if I register a domain and have developed it into a complete website. From where and how Googlebot knows that the new domain is up? Does it always start with the domain registry?
If it starts with the registry, does that mean that anyone can have complete access to the registry's database? Thanks for any insight.
Google needs to find you. That is, if there is no link to your site from another web site, it'll never find it.
Google finds pages to crawl as links from other pages. If no site links to your site, Google will likely never find it.
Google will find your website on its own if some existing website has a link to it.
You can jump-start the process: http://www.google.com/addurl/.
You may also be interested in Google's Webmaster Tools.
Usually I submit new domain directly to Google, I found out it takes less time for Google crwaler to find out about it rather than waiting for Googelbot to parse some link to the new domain in some other websites: http://www.google.com/addurl/
Another way is to create a free Webmaster Tools account on Google and add the new domain/website there (maybe also with a sitemap), this will take to Goggle even less time to crawl your domain and get it indexed.
As far as I know the registry's informations of a domain for most domains are public, you can read them by using a simple whois service.
You should look at the google website. They have some good information here. They even have a link to add your site to their list to crawl.