Do you want instructions on how to buy a domain name, or are you looking for advice on web hosts?
It sounds like you want to at least purchase a domain name. Once you have a domain name, there are a few options. You can go with a hosting provider and set up an account with the domain name pointing to it, or you could continue with your current hosting setup (presumably) and pay a small fee to your registrar to redirect to a different URL for you.
If everything is OK with the site you are at, and you just want a different URL for advertising, it might be easiest to stay where you are and pay for a redirect on the domain name that you purchase. Hosting solutions will provide you with more space and more flexibility, but if you just want to run a blog then this might be overkill.
Willful ignorance of science is not commendable. Refusing to learn the difference between a credible source and a shill is criminally stupid.
Bring it down to a minimum number of components to boot it. I'd try taking everything off of the system except for 1 stick of RAM, CPU, Video Card, and CD-ROM. See if you can boot it off of a bootable CD-ROM.