The built in search page has a number of configurations...
First of all, look at the permissions.. Advanced search is not active by default.
You can see how far the indexation reach. New added content is not yet indexed and therefore not visible in a search. You need to wait for the cron to do the indexing.
Here you can set the minimum number of characters the search needs. You also can log the search. Usefull but a little bit slowing down your site.
By deafult there are 2 search pages present. One for content and the other for the users. With content you can give a weight to the search. E.g. if found then the new content will be on the top.
If you search for a word...
If the permissions are good, you can use the advanced search and search more specific.