H! My name is Miroslav. I'm working in IT field over 15 years. I have experience with Joomla upgrade and migration between servers, have done it before. As you can see in my milestone, I have planned migration:
1) Check all necessary settings on the old and new server. Make a backup of working Joomla site before starting anything.
2) Upgrade Joomla from 2.5 to 3.x and then to the latest version, now is 3.8.12. Restore Joomla on a new server and upgrade version of PHP. Upgrade all necessary components, plugins, and modules.
3)Check every page in 2 different browsers, if everything working fine. Fix bugs if I found some.
I need from you information about the current template. I see this as a potential problem, because the template is written for 2.5 version, and could cause some errors after migration. Also, I would like to know more about your server/hosting provider.
If you have any question for me, be free to ask.