When I say "start from scratch", I mean work with a bare-bones framework and build on it. Frameworks are basically minimal themes that are meant to be used as a basis for other designs. They are stripped back in terms of CSS, but provide a lot of functionality and "hooks" built in. You basically create a child theme that uses the framework as a starting point for functionality. All the basic elements are there, navigation, sidebar, footer, etc, you just add your own CSS/HTML/whatever on top of the framework to make it your own.
Some frameworks to look at –
Thematic (free),
Thesis (premium),
Genesis (premium)
Whether you use an existing theme and customise it, or make use of a framework, you should always create a child theme and never edit the code of the parent theme. Alternatively, look at the
ModX CMS as a (better) alternative to Wordpress.