I'm gungho to get started building in Craft but I don't want the complications of moving the site over from a development directory. Is there a way to easily do this while the current site is up and running?
In general, developing a website on the live server is a risky thing. Should something go wrong while you're developing then you risk the live website. It's unlikely, but an accidental upload to the wrong folder, or a wonky SQL statement, or anything like that can happen.
If you must run the site on the live server though, your best bet is to set up a sub-domain such as beta.mywebsite.com. Host that on your live server and do all the development there. Then, when it's time to retire the old site and make the new one 'live', you can move the files across from the beta directory to the www one.
I wouldn't recommend doing that though. Much less risky to get your local machine set up as a development server, do it all on that, and then put the site live.