Testing updates to an existing site on a staging server

I have a WordPress site and would like to create a copy of it on a different server so that I can test updates to it and then deploy the updates to my production server. Is there a clean/simple way to do this?

I’ve tried a couple of methods already:

1. Copy all files to the different server and change the URL in the admin interface to the new server. The copy of the site was identical to production, but seems to still be using my production database for the copy of the site. Also, I can’t just copy all the files back to production after making the updates (e.g., adding new pages), for fear of overriding changes that have happened in production while I was testing.

2. Create a separate WordPress installation on the test server, copy the wp-content folder over to it, and export the data from production / import it to testing. This did not work well at all, because the look and feel of the site, as well as some of its functionality was not reflected on the test server.

I would really appreciate some advice on a better way to do this! Thanks in advance!