Submitting your URL to Google is simple. Firstly head over Webmaster Tools > Submit URL.
Type your website into the URL: box, fill out the RECAPTCHA box and click Submit Request and you will see this below if its successful.
Now give Google time to process this and your website will be on Google.
As your website is new you will not rank very high at the start so if you would like to check Google now and then to see if its been processed you can simply type into Googles search bar site:yourdomain.com
Want to submit your URL to all major search engines in one go? Head over to www.entireweb.com and simply add your website and email address, select a plan (free), you will receive an email to confirm your submission and then Entireweb will proceed to submit your website to multiple search engines.