Contact the owner. Before so much as hinting at a price, simply email to ask whether or not the domain is for sale. If you are known or can clearly be linked to a thriving business, create a generic alternate email address through which to contact them, as your success might be leveraged against you. Be aware, however, that an informal-sounding email address is more likely to be regarded as spam or junk mail.
In housing terms, VPS hosting is like renting your own apartment in a larger building. You're much more isolated than in the roommate situation mentioned above; it's still possible that a neighboring apartment could causes annoyance for you, but far less likely. In web hosting terms, Site A's traffic surge won't have nearly as much impact on Site B or Site C. As you'd expect, VPS hosting costs more than shared hosting. You'll pay roughly $20 to $60 per month.
Usually a single machine placed in a private residence can be used to host one or more web sites from a usually consumer-grade broadband connection. These can be purpose-built machines or more commonly old PCs. Some ISPs actively attempt to block home servers by disallowing incoming requests to TCP port 80 of the user's connection and by refusing to provide static IP addresses. A common way to attain a reliable DNS host name is by creating an account with a dynamic DNS service. A dynamic DNS service will automatically change the IP address that a URL points to when the IP address changes.