The look of the default results page is indeed printed from the search.php in your theme directory. It’s true for the most themes, some others might use different files.
If you decide to customize that page, I recommend creating a child theme, so you won’t loose your modifications on updates.
Unfortunately I cannot provide an exact tutorial on how to customize that file, as it’s very different for each and every theme out there. If you are unsure of some things, I highly recommend asking the theme author for advice about modifications. They usually tend to be very helpful.
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