Hi there! I don't know how much you are willing to spend here, but a simple VPS shouldn't cost you that much. For instance you could check
https://www.transip.eu/vps/pricing-and-purchase/ for great Holland-based VPS services or
http://www.hostgator.com/ for an US-based one.
VPS management is usually easy, as providers try to make customers feel as cozy as possible. There might be some general rules/tips on VPS management, but I'm sure your provider of choice will be able to inform you about everything you need to know (it's their job anyway
)
Next, you can't really operate a domain - that's just and address which you are able to point wherever you want to. So, say you purchase a VPS plan and a domain (
http://www.godaddy.com excels at it), you can link your domain with your VPS's IP address so that people can query your server (web, mail, whatever) through the domain you just purchased. That should be also quite easy to set up, specially since both the domain name provider AND the VPS provider are surely able to give you instructions on how to set it up properly for your particular requirements.
Finally, .onion domains are quite different from regular ones. They're free to get and manage, but a little difficult to set up. I'd suggest you read the Tor project documentation here
https://www.torproject.org/docs/tor-hidden-service.html.en. If you want to host a Tor Hidden Service (i.e. a website within the Tor network) you can either use your own machine or a VPS as the host computer. I'd advise you to stay away from APACHE server software if you want your website to be secure and really hidden, since APACHE is a large chunk of code with many possible holes/vulnerabilities. Instead, use something like Savant or thttpd. The rest of the info is well explained at the TOR documentation linked above.
If you still need help on setting up the VPS/domain/server/savant/TOR/ whatever just PM me for more details. I'd be great to know a) whether you have Tor/Tor bundle browser or anything like that installed and b)your main OS.
Hope my help is useful, and good luck on your project
Cheers!