Self-Install vs. Hosted
You can get a free WordPress site at wordpress.com, but if you're doing serious work on the web, that's probably not the way you want to go. Sure, it's super-easy, but because the folks at WordPress control your installation, you'll find that your options are limited. Want to install a cool theme you found somewhere online? You can't do that. Want to enhance the functionality of your site with a plugin someone recommended? You can't do that. Want to modify and tweak your site's design? You'll have limited options. WordPress.com "owns" your installation, and you'll need to play by their rules.
If you want to run a professional web site, you'll need to install WordPress software on your own domain, in a proper web hosting account that's fully owned and controlled by you.
The WordPress software you're allowed to download and install on your own server is available at wordpress.org (not .com). Same organization, two different domains, two different use cases.
Fortunately, most web hosts make it trivially easy to install the software available from wordpress.org directly into your hosting account with a few clicks.
The first thing you need to do is to purchase a standard web hosting account. There are literally thousands of choices out there, but take note: Not all web hosts are created equal! They may look similar on the surface, but there's more to choosing a web host than comparing prices and features - service and support, security, backup systems, and system monitoring really do matter!
Most people will only ever use a tiny fraction of the features offered by a web host, so don't let things like "unlimited email addresses" or "unlimited bandwidth" be your main criteria. Instead, do a search for "web hosts compared" or "recommended web hosts." As with most things in life, you'll probably be sorry if you choose the cheapest host you find. In our experience, people have the best experience with mid-priced hosts that use the cPanel hosting system. We like bluehost.com and justhost.com, though cPanel is used very widely.
The rest of this tutorial assumes that you're set up with a decent web host and that your domain resolves in a web browser.