The best way to get your site listed quickly on Google is to submit your sitemap in Google's Search Console. This is a simple process but the following steps need to be completed in the order listed below:
Submit your Sitemap to Google via Google Search Console
If you haven’t completed steps 1 and 2 above, please refer to this help guide which explains how to setup Google Search Console and verify your website. You can then return to this guide and follow the process below.
Now you're ready to submit your sitemap. Please log into your Google Search Console account and follow the steps below:
Expand the "Index" menu on the left menu
Enter sitemap.xml in the "Add a new sitemap" box after your verified domain name
Google will now place your sitemap in it's queue and when you reach the top it's Googlebot spider will visit the sitemap and index your pages. Once it has done so you'll be able to see how many pages are appearing in the index and whether it found any problems while spidering the site.
We recommend that each time you make changes to your website in Create and publish, that you also re-submit your sitemap in Google Search Console too.