Before attempting to upgrade, you should always take a backup of your site. Akeeba backup is great for this, as it allows you to backup to the cloud, download, or store the backup on your server.
You can get akeeba backup here: https://www.akeebabackup.com/download/akeeba-backup.html
This will install like any component you've installed before, through the Extensions menu, then Install/Uninstall. Once this is installed, configure it using either the configuration wizard, or the manual configuration option, then hit the backup now button.
Once the backup is complete, we can begin to upgrade.
You'll need a component called jupgrade. This will update your core Joomla files to the latest version (As of writing this, 1.7.0). You can get jupgrade here: http://www.matware.com.ar/downloads/joomla/jupgrade.html
Install this in exactly the same way as you did with the akeeba backup file. The jupgrade option should now appear in the Components menu, click this to take you to the upgrade page.
Above the big green "Start Upgrade" button, you may see a line of text stating that "Mootools 1.2 not loaded", if not, skip to the next paragraph. If this is the case, click the text displayed and it will take you to the Components manager. Type in the search bar "System – Mootools Upgrade" and you should be presented with a single component. Click on the red X icon about halfway along to enable it, then return to the jupgrade menu with the big green button.
Click on the big green "Start Upgrade" button, and jupgrade will begin to upgrade your Joomla! Website automatically.
If all goes well, you should be presented with this.
It says here that your site has been upgraded to Joomal! 1.6.x, but in fact, it's been upgraded to 1.7.x.
Click either link to be taken to your new, 1.7.x Joomla! Website, or the Admin panel.
If anything during the upgrade did not go to plan, you can simply restore the backup you made using akeeba backup, and try again!