There are certain steps of updating drupal to new version .
I am here sharing my experience .
1) Please take the database backup of your drupal website from phpmyadmin.
2) Please backup your theme and sites folder .
3) Please deactivate all the modules except the core modules .
4) Please turn on the maintenance mode from admin panel.
5) Change the theme to drupal core theme.
6) Unzip the latest drupal version and upload it to your server.Please upload all the files/folders accept sites folder.
7) When uploading is done just run http://yoursite.com/update.php .
8) Now upload the concern theme to your theme directory and then change the theme from admin panel .Please update your theme .info file as per the drupal version otherwise you will get errors.
9) Now activate all the modules except the core ones which you have deactivated earlier.
10) Now you are done with this.
Please share your experience if you have encountered any issue.