I've bought a virtual server hosting account to play with and eventually move this website onto (it should work out cheaper and give me a few more options). Bit of a steep learning curve though as it's all linux and ssh stuff and so completely over my head at the moment. I do feel like a proper geek logging into the server and getting a command line though. A bit different from remote desktop-ing into Windows servers like I'm always doing at work.
For geek-cred, I can now point you at my own IP address: 18.104.22.168. Unfortunately, it's only showing the default Apache test page at the moment. I haven't actually managed to get the server to do anything yet.