It depends on what you wanna do. :o
In any case, you have to buy a domain name, which costs around $9~10/yr. One thing about domains, though, is that you need to enter your real contact info, and it will be publicly available to anyone who does a whois record request on your domain. This was a great idea back when the Internet wasn't full of questionable people. There are whois privacy services, however, which basically act like proxies and you can fill out their contact info in your domain instead. Those cost like $1/yr or something, and some domain registrars bundle it with the domain name service itself.
I've been using
Namecheap and I've had no complaints. GoDaddy is evil, avoid 'em. As for the others, I dunno. I believe
eNom is a pretty well known registrar, too. Look around and compare prices, especially taking note of renewal costs. I remember some registrars giving you a good deal on the first year, but then renewal for the next year is higher, etc.
Once you have a domain, you could actually link it to a Blogspot blog to make it look more professional, if you like the features it provides. Or you could link your domain to a Google Apps account, which would give you access to Google Sites, email for your domain, etc.
If you want to run Wordpress, though.. I think there is a free Wordpress hosting service that can be linked to a domain as well?
If not, then you will have to buy web hosting which for a small site will cost about $5/mo. Then you'd install Wordpress on there, along with whatever theme you want etc.