Hi Gijs,
Moving the search box is actually to change the layout. Changing the page layout is available. This can be achieved via editing the master page. Using SharePoint Designer 2013 is suggested. The master page determines the layout. If you want to add a new button to start workflow, you can edit current page to design this button, such as developing.
About developing the design and customizing page layouts, here are some articles for your reference:
http://msdn.microsoft.com/en-us/library/jj733518.aspx
http://msdn.microsoft.com/en-us/library/dn205271.aspx
Thanks,
Kevin Cui