Then you would buy a domain, get a webhost (most domain sites also host, but you could use different providers if you want), and then install Wordpress onto your webhost account. Then you will be able to use Wordpress (which is all kinds of awesome) to control your blog and the URL will be yourblogname.com. I'm not an expert, but I could try to answer general questions you might have.

I'm not sure if the process would be the same for Tumblr.