In a page template you use the template tag [-- SEARCH_FORM --]
If you don't like the way the default formatting for the form is you can use that tag to display the required html on a page and then copy that html and modify it and replace the [-- SEARCH_FORM --] tag with your modified html.
ON the search results page template you can use the tag
[-- SEARCHRESULT_FORM --]
Again if you don't like the format you can copy the html generated by that tag and modify it and replace the tag with your version of the html.
Note: in order for the store to be indexed for searching you must have at least 1 page which has the page field Product Search Field: check box checked.
These 2 links give more information on the above tags.
http://www.shopsite.com/help/8.3/en-US/ ... -page.html
http://www.shopsite.com/help/8.3/en-US/ ... earch.html