What’s the best way to submit a website to search engines?

Some search engines allow you to buy placement. Most don’t however, usually most search engines just search your site for keywords, the more keywords that match someone’s search, the higher in the results you are. If there’s a tie they usually go and look at backlinking which means “which site has more votes?” such as Google. Google (the biggest search service there is) uses a voting system. If you link to a site, that counts as a vote. The more votes a site has the higher it ranks in the results. So if you want to go up in the results, get people to link to your site!