Rans6Andrew,
I'd suggest first thing is to look at who you can get the domain from as they may well be able to provide space.
When I put up Mrs batninth's web site, we get space from our ISP, Virgin media, so the site is on the Virgin media space, and I just redirected the domain name to the site.
In your case, it looks like you need the space so I'd say look around and see if you can get the domain plus the space you need hosted somewhere for a reasonable price
Either way, when you sign on for the domain and the hosting, they will give the instructions of what to do, ids, passwords etc. Generally you are right in that you need to copy all the files to the site, you can do it with Internet Explorer but I use FileZilla which is an FTP client (I'm a tight so-and-so and FileZilla is free!) to upload everything.
So - sign up for space & the domain you need, they will tell you what to do, and get an FTP client loaded.
Frankly if I can do it then anyone can. It's easy to do & you'll have no problems