WordPress is a great software and gives you two options for using it.
First, you can always set up a free WordPress at WordPress.com. You don't need to know much about technology or how to get it going, and you'll be blogging in minutes.
Second, the full version of WordPress can be found at WordPress.org. This is the version you can customize and edit the code to your own desires.
You need to download the newest version, unzip it, and then publish it to your hosting account by FTP. There are some things you need to check; your hosting needs to be running MySQL and PHP, you want to double check your versions to the current builds requirements. This hosting company I know works, http://www.redfusionhosting.com.
Once the site is posted, you need to create a database. Most server management tools, (like CPanel) allow you to do this quite quick. Write down your username and password for the database you set up, you need it when you go through the install process. You'll need to edit the install page, adding your settings to the code with a tool like Dreamweaver, or even Note Pad, so that installation will be able to be completed.
Now, login to you home page of the domain you set up, it will ask you a couple of questions, it will go through the install process, and you're done. Make note of your password, is always something too hard to remember, you can change it later.
Step by step instructions can be found at WordPress.org.