Having trouble moving my site URL

The situation:
One of my domains expired on Godaddy while I was holiday. I refuse to pay the $80 fee they charge, and bought another domain since it was’t a very high traffic site anyway. So I’m trying to change my WordPress blog’s domain, and not the hosting provider.

What I’d like to do
I’d like to have the files of the directory as is, and point the new domain to the files. In theory, doesn’t this mean since I installed WordPress in that directory? Or is there something that I’m missing here?

I’ve tried adding the new domain and pointing it to the already-present directory, but all it does is display it as plain HTML without images (basically not reading from any themes, css files etc and not accessing the images directory. I think it might be because the URLs for this still points to the older domain that I no longer exist.

Is there an easy fix for this at all? Or do I have to make a new directory, install WordPress on there, then move my files across? Could someone point me to a good tutorial about doing this?