Setting up a WordPress staging environment?

I have now installed (on my own server) and am hosting my site via WordPress (again, on my own server). Phase 1 of the site is now up and running live and I would like to begin development of Phase 2 of the site (modifying some templates via a child theme, adding new sections via these new templates, etc.).

I’m wondering if anyone has a suggestion on the best approach for this? Should I clone my live site to a development/staging directory? What’s the best way to go about doing this?

And then, once I’ve completed Phase 2, what’s the best way to “push it” live?

Any advice would be greatly appreciated.

Thanks,
Kristin.