Can I "cut and paste" the whole site into a new programme and do it from there?
Pretty much, yes. It's a simple site. I assume you want a free solution, so something like BlueGriffon might work for you. Since you've previously worked in a vaguely wordprocessor-like environment, OpenOffice/LibreOffice would also work.
Neither of the above would give you a particularly polished-looking site.
However, your current host offers several website building tools, so I suggest you talk to it or have a look around its website.