Going from locally testing to online server

After testing locally my first WP site I need to go online to check how it will perform in the real world.

So I need some advices.

Is there anything special or different from local installation to pay attention to?

I need to test it online also to show work progress to customer, what’s best way to do that? How to put WP in a special directory visible only if you know it?

And what about moving site to public directory when ready?

Thanks in advance for your help.