You're running on Joomla 2.5 on your site. You're going to do the mini-migration to 3.5.1. It will force you to stop there and re-login. You will then update to 3.6.5 where it will stop there and force you to login again. Then, you'll update to 3.10.x. Finally, you'll do the mini-migration to 4.x.
Most likely, you will need to change out your template as it won't be compatible with Joomla 2.5, 3, and 4. Before you start, change the default template to Beez3 in Joomla 2.5. That template should get you all the way through to Joomla 4.x. Then you can either use Cassiopeia or a different template in Joomla 4.
When it comes to testing in between all the steps, do it. However you won't really be able to see the frontend unless you set up all the module instances in Beez3. To do that, type example.com/?tp=1 to find all the module positions and assign your modules to them. This is only so that you can test functionality. Styling won't matter until you get the template you want to keep installed in Joomla 4.
Test the backend thoroughly. Make sure you can add/edit articles, modules, menu items, etc. Do this when you get to 3.5.1, 3.6.5, 3.10.x, and 4.x.
Run the DB Fix in Extensions -> Manage. Clear Joomla Cache (System -> Clear Cache) between updates. Do backups between each step!!
Check all extensions at the beginning. You might be better off uninstalling whatever you can and reinstalling in 4.x if they're easy plugins/extensions. Alternatively you could use an extension like SPTransfer or J2XML. Create a Joomla 4 site and bring over your content.
Once you get to 3.10.x, follow the instruction in the Step by Step 3.10.x to 4.x document including the pre-update check. make sure you remove leftover 2.5 files that might be lingering.