You can do both. How depends on the browser you're using.
Most have a menu setting for Tools and maybe Options. There's generally a setting for what homepage to use and if you're on the one you want at the time most allow a "Use current" button that lets you say "make the page I'm viewing now my homepage".
Setting a desktop shortcut is as easy as dragging the browser icon from the address bar onto your desktop.
Deb, in New Zealand