It's worth wondering whether you need to submit your newly made website to search engines in order for them to know its presence.
For a quick answer, it's a no. You don't need to submit your website manually to search engines because they have evolved to become better, making that task not anymore necessary. There was a time when you manual submission was recommended and worth doing.
But not anymore.
Search engines use crawlers to fetch your website and see what it has. So as long as your website follows search engines' guidelines and doesn't do manipulative things or do unethical practices, for no fee, your website should be included in search engines' results relatively fast.
The only thing you need to do, is have your website live and running, and accessible by typing the domain name directly into the address bar. Another thing you should consider is to allow search engines to crawl your website via directives on your website's robots.txt.
After a few days to a few weeks, your website should already be indexed and available on major search engines, such as Google and Bing. This is because they "submissions" happen automatically.
But if patience is not on your list, or if you just want to speed up the process, there are a few things you can do:
Webmasters Tools
Both Google and Bing recommend that you have a Webmaster accounts with them. With you having accounts on both Google and Bing, the two dominant search engines of the web can communicate with you if they see any problems concerning your website's crawlability. They can also give recommendations about how to improve your site's visibility.
By having your website registered to them, you're plainly submitting your website to them.
Backlinks
When you created a website using a new domain name, there won't be anyone on the web that clearly link back to you. Having a link to your website from another website will make search engines find your website faster. If you can have a backlink pointing to your site from a highly reputable, well-established and regularly updated website, the faster the search engines will find your website.
The fastest and also the easiest way to get backlink, is to comment on a newly published article on a popular blog. Most blogs allow visitors to put a link back to their own website when posting a comment.
Social media
Facebook and other social media allows anyone to register for free. Having an account there allows you to put a link back to your website. Social media networks in general, have a high visibility on search engines because posts and profiles are indexed quite often, especially where they're active.
Another good thing about social media is that Google also has Google+ where you can create a page for your business. Having a page there will benefit as it can build both trust and authority with Google. You can also link your business's Google+ page to your own website. And because Google+ is Google's, your website's link will be instantly crawled and indexed by Google.
Alternatively, if you have a personal Google account, you can "share" your website's URL. And if you have integrated social media buttons on your website, you can simply click the +1 button to instantly tell Google to index your website.
Crawled Vs. Ranked
While being indexed is relatively easy, sometimes there are confusion where people think that their website will show up on search engines' results page after typing for specific keywords.
This has nothing to do with search engine submission. You can't submit your website to search engines and tell them that you want to be ranked for some keywords. Gaining top position in search engines' results page is another feat, and isn't at all the same thing.
Unless you're willing to pay some money on search engine marketing (paid advertising), the only sure way to gain good visibility and rank is through development, SEO (organic means) and others. You need to target the correct audience that will love your posts, you need to create plenty of contents, posts, and so forth - ensuring that they contain the keywords/phrases you're targeting.
By carefully targeting keywords, you can ensure your website to get ranked dependently on search engines' algorithms, relevant to your authority and credibility.