Indexing is fully automatic, you only need to run it manually once.
I would recommend a manual re-index every now and then, maybe twice a year or so depending on your site, as some rare things aren't updated automatically (if you delete a tag, for example, it remains in the database).
Default search in WP doesn't use indexes, but queries the database directly. It's fast, unfortunately it's also completely unable to sort results by relevancy.