Just enter your website name in Google and the result will clear your question.As far as any search engine is considered the actual listing of your domain, once it goes live is...it can take a few hours to maybe 1-2 days. This really depends on what you do yourself for the search engine to find your website, crawl it and list it.
You need to let know the search engines that you exist which you can do my marketing.For example, let’s say you decide to launch your website right now and your website is live but it’s brand new. So, for publicity/marketing you can add in social networking sites like facebook,twitter etc about your website and add a link.People visit it and the search engines crawl your link and thus you get indexed.
Another option you can do is create a Google Webmaster Tools account and there is a feature there, that allows you to send the Google bot to fetch your website thus indexing it. You can refer www.google.com/webmasters/tools/ for more details .
Another way is follow techniques like guest blogging and using your website as a signature,publish articles relating to your website,proper SEO techniques,marketing/advertising etc.I suggest you to use the SEV(Search Engine Visibility) from http://www.thewebpole.com/search-engine/seo-services2.aspx?prog_id=456994&pl_id=456994
as they cover many techniques like keyword and suggested topics,submit the site to over 100 search engines,submission to directories,sitemap creation and also provide reports which will eventually improve your web presence and indexed soon.