There are several sites that offers services to register your URL to several search engines but I think that registering it to Google and to Bing, that are the biggest ones, is more than you need … and this can be done simply without using third party services that always ask you email and more.
Therefore to register an URL to Google search engine, you can go here.
You can even ask to re-index some specific URL that possibly changed: it will take time but the position and priority it will be increased by the search engine:
To register in Bing you can go here.
After few click you’ll receive the following notification in both cases: “Your request was successfully submitted!”
Update 2019 – Now you have to use the Bing webmaster tools site in order to add (for free after logging into that site with your Microsoft account) sites and specific URL you want to submit/re-submit to Bing:
You may need to wait possibly few days before you new URL will be registered, after a verification done by the search engine … from then on, it can be found searching it!