Here is Joomla 2.5 to 3.5 migration tutorial: https://docs.joomla.org/Joomla_2.5_to_3.x_Step_by_Step_Migration

It says one need to:

0. create a full backup of your web hosting account and verify backup is usable

1. check web hosting software if it meets new Joomla requirements: https://downloads.joomla.org/technical-requirements

2. write down all extensions/components/modules and check if they support new Joomla version.

3. create new subdomain on your hosting and new mysql database into which you copy your joomla live website and do a upgrade on this new testing installation. Before proceed, modify configuration.php to match new path and new mysql credentials!! Else you may mess your live installation.

FROM NOW ON MADE ALL THINGS ON TESTING INSTALLATION, NOT ON LIVE SITE:

4. Login administration and disable all third party modules/extensions/plugins.

5. Download latest 2.x version of the Joomla from https://downloads.joomla.org/cms/joomla25 , download "patch" kind of package, not "full". In my case latest was v. 2.5.8: https://downloads.joomla.org/cms/joomla25/2-5-28/joomla_2-5-x_to_2-5-28-stable-patch_package-zip?format=zip

6. Unpack downloaded zip archive files and overwrite all existing files in your Joomla testing installation (on a subdomain) by them. See quickly if site loading ok and admin area.

7. Download "patch"/"update" kind of package to version 3.5.1 (newer 3.x seems not to support patches from v. 2.x): https://downloads.joomla.org/cms/joomla3/3-5-1
Again extract it and overwrite your testing installation files by ones in the downloaded zip archive

8. Login testing installation administrator area and try to enable plugins/extensions/modules one by one - after enabling one, try to open your site front end/articles.. to verify there are no error. If facing an error, you need to disable that particular extension (https://www.ostraining.com/blog/joomla/disable-a-joomla-plugin/) and find update for it.

9. upgrade your live site same way as a test site