Unfortunately moving between major versions such as from 5 to 6 or from 6 to 7 is a much bigger topic and too large for a single tutorial.
Before you start, please remember one thing: backup, backup, backup, backup, backup, backup, backup, backup your site!
Step 1: Check for Available Updates
To get started with updating your Drupal site, click on "Configuration" and then "Available updates".
This page will show you an updates you need to make. They will be marked in red as in the image above. If there are updates, it is worth putting your site in maintenance mode at this point to avoid any problems for your users as you update. Instructions for doing that are here.
Step 2: Download the Update Files
Drupal will show you a link to download the files for the new update.
Step 3: Upload the Update Files
Unzip the files that you've just downloaded.
IMPORTANT: Delete the /sites/ folder from these unzipped files. Do not upload the /sites/ folder as this will overwrite any custom work that you have done. Make sure that you leave the exisiting /sites/ folder intact on the server.
Then open up your FTP program or file manager and navigate to your existing Drupal site files. Simply upload the new files over the top of the existing ones.
Once the upload is complete, you can refresh your "Available updates" page and hopefully it will now be marked in green rather than red.
Step 4: Run the Update Manager
Click on "Modules" and then click on "Always run the update script each time a module is updated".
Click on the "Administration pages" link and go "Configuration" then "Maintenance mode" to put the site back online.
