First of all, go to the Submit URL – Google site. Enter your website’s URL, and prove that you are not a robot
Once done, you can submit your website to be indexed by Google.
A short message should confirm it.
After that, if you try to look up your site, you might see that Google doesn’t see it as completely indexed – you have asked it to index it, but the website ownership hasn’t been claimed yet – let’s do it by clicking on the link.
On the page to which you’ll be redirected (you might have to log in / register first), click on ADD A PROPERTY
Here, enter your website’s URL, and validate by clicking on Add
The website will be added to the list of properties. In the Manage property dropdown menu, click on Verify this property
You’ll get here several methods of website verification – the Alternate methods are the easier ones. If you have setup a Google Analytics website (see Google Analytics how to add a website to your account and get a Tracking ID), all you have to do is select the first option.
In any case, select your favorite method, do the required steps – once done and verified, you should get a confirmation message
There are more steps to get your website properly indexed by Google : adding a robots and sitemap files, verifying dead links, …
However, these few steps should get you quickly started without much effort.