There are places advertising free domain name registration. There are some simple things you can do to increase the likelihood. First, give each of your web pages a good, descriptive title. The title is a page property, not text on your web page. It doesn't need to be short.
Many search engines start with the page title.
Be sure it has your business name and some good descriptive words. The title also is what shows up in your Bookmarks or Favorites. Rather than assuming the search engines will discover your site, register your site with popular search engines. There are services that will do this for a fee. There are also services that will do some for free.