I only want to display a subset of the buttons on the html editor button bar. What is the proper way to remove some of the quicktags? For intance I don't want it to offer "proofread".
You can use the TinyMCE Advanced Plugin that let you configure the Wordpress editor, removing the undesidered buttons.
Install it, via Plugins -> Add New
EDIT: For HTML Editor, try to use the HTML Editor Reloaded plugin.