Well, the first and the foremost solution or the approach you can have is by visiting the Joomla document for upgrading Joomla website from version 2.5.x to Joomla 3.6.x. You can use the below link for the same:
Today, I had performed a similar task and will be sharing with you the exact approach which I had used for upgrading my website.
Note*: Take a complete back up of all website files and the database. Keep the files secure offline.
—BACK UP ALL FILES AND DATABASE—-
Now, leave the current website live with old Joomla version and configure a staging site on a sub domain with your back up files, please use a new database for this purpose. You can even use another hosting or even your local machine for this job.
—Configure Staging Site i.e. Replica of current website with Joomla 2.5.x Version—
Once you are done setting up the live version of website on staging mode, now the time has come to proceed with the upgrade process.
—Install & Configure Akeeba – A backup utility tool for Joomla—
PHP Version 5.3.3+ and Joomla 2.5.5+
Compatibility check – PHP Version & Joomla
Once you’re done with the set up, please take a back up again.
—Download & Install Joomla 3.0 upgrade package—
If any errors or issues occur you check the components or extensions in use. Here, you can simply take a note and disable all of the concerned extensions or plugins till the Joomla gets upgraded.
If the above installation is successful, you can purge the cache, fix database (under extensions), etc. to ensure the upgrade is in effect.
Or in case, you find any issues in the default admin templates or backend after installation, you can simply extract Joomla 3.0 full stable release and use it to replace or overwrite all the previous files on the staging.
Now, Purge the cache, fix the database and see if everything works fine.
—Now, you must be getting an auto upgrade Joomla Installer option in the admin menu, this option can be used to upgrade Joomla to its latest Joomla 3.6 version—
—If installer is not working, you can simply download the Joomla 3.6 upgrade package and install it like you installed the version 3.0—
Once, you are done, please purge all cache and fix the database (under extensions)
If there is any technical issue or some kind of errors, you can simply download the Joomla 3.6.4 full stable version and use it to replace or overwrite all the previous files on the staging and I believe this will resolve your concerns.
—Now, the time is to make your website fully functional with latest Joomla 3.6.x—
—Enable all components or plugins if any disabled earlier and upgrade it to the latest version one by one—
If you find any problem with the Joomla installer, make sure the Installer plugins are enabled.
Make sure to take back ups at regular intervals to avoid any work loss.
Also, please clear/purge cache & fix database in case of any new upgrades or updates.
Throughout this upgrade process you might come across different issues or errors, don’t panic and loose your focus. These issues can be fixed. Make sure you have backs ups ready at every instance and you are working on your local machine or some sort of staging website to ensure your live website is unaffected.