I'm glad you decided to do it this way. At the very least, the experience will be beneficial to you in the future.
1. The first thing you should do is download and install an FTP client. This makes the upload process much simpler and easier. I recommend Filezilla. You may or may not end up using Filezilla in the future (though I suspect you will) so I will not go into much detail on setting it up. At the top of the screen you will see four entry-boxes followed by a button that says "Quickconnect". The Host is the URL or IP address of your website. The username and password provided to you by your hosting provider for FTP access. The Port can usually be left blank, unless your host has set a specific port to connect on (21 is default). Click "Quickconnect" and the files of your website will be displayed on the right-hand side. The left side is for browsing your computer.
2. Download Joomla!
3. Unzip the Joomla! files into their own folder somewhere on your computer that is easy to browse to.
4. Using Filezilla, browse to that folder.
5. Highlight the contents of the folder and drag all of it "onto" your website. This will start the upload of Joomla! to your web server. It may take awhile to complete, so be patient.
6. Once all the files are uploaded, go to http://your-site-here.com. This will load Joomla!'s built-in web installation.