Getting Your Hands Dirty
If you're a writer or buyer, and don't care too much for the nitty gritty, it's okay! WordPress.com is an easy way to get started blogging. There you can sign up for an account, get a free blog (i.e.- YourBlogTitle.Wordpress.com). Follow the directions in WordPress.com's Getting Started Guide and you'll be blogging in no time.
For developers there is WordPress.ORG (sorry to use caps, but it's an important distinction). WordPress.org is where you can find information on downloading and installing WordPress. The WordPress.org Installation Guide will walk you through downloading the package, uploading it to your own server, creating a mySQL database, and running the WordPress installer.
Also, ProBlogger has a great screencast on the whole process.
Okay, maybe you're thinking, "but I just want to tinker around, and I don't have a dedicated server for development." Yes, you do (hint: you're looking at it) and if you aren't developing on your own local server, then you are missing out! Working this way allows you to develop offline, and it's fast. However, I do not recommend using the software below to host your own website from home. Developing with minimal security can be dangerous online, and working locally reduces the hassle of securing the site during development. Just remember that these servers are designed for developing - not hosting.
If you're on a Linux box, Supriyadi Slamet Widodo's blog has an excellent guide to setting up a LAMP server on Ubuntu, and instructions on installing WordPress on your LAMP server.
One of the most requested WordPress related jobs is Theme Development. WordPress's Theme Development documentation is fortunately thorough and relatively easy to read.
There you can get started on learning how to develop a theme from scratch, or get resources on how to modify an existing theme. If you are modifying themes, be sure to appropriately credit the original designer, even if your theme looks completely different, the time spent developing the back end deserves credit, and it's the right thing to do.