You don’t need to worry about submitting to lots of search engines. The most important one is Google.
Once you have submitted your site to Google, it’s a good idea to also submit it to Bing. This helps Yahoo and DuckDuckGo find your site too.
The other search engines you should consider submitting to are local ones. For instance, you could submit your website to Yandex if you want to get more visitors from Russia.