How to change Parent Themes in Child Theme CSS

I think you are confused about this. Let me explain.

The @import url("../responsive/style.css"); is telling the Child theme to import the Responsive themes style.css. This is important because without this the Child theme doesn’t know what its Parent theme is. So the @import url("../responsive/style.css"); points the Child theme to the default (main) file in the Responsive theme (in most all WordPress themes the default or rather main file is the style.css which includes the themes header defining what it is).

So basically without the import function in the style.css of your Child theme, the Child theme would be broken and there would be no link between the Child theme and Parent theme.

So how do you find the path to put in the import function (i.e. /responsive/style.css). You can find this by entering your FTP (FileZilla, cpanel, or whatever client you want) and going to the public_html – wp-content – themes section. Then find the theme folder of the theme that you are going to make a Child theme for. Put that folders name in the import followed by the style.css (i.e. /thethemefoldername/style.css).

Sorry if that sounds confusing. So basically you need to figure out what your themes file name is that you are switching to and the replace that with the responsive in the import function. That will point that Child theme to the new theme that you desirer. If you change that then that is all you need to do.

You can just use a Child theme plugin. I would recommend https://wordpress.org/plugins/child-theme-configurator/ or https://wordpress.org/plugins/one-click-child-theme/ . With these you can just create a new Child theme and then add your new stuff to that.

And how exactly do I create a backup in case it gets messed up when switching over?

You should always make backups of your website. There are a few ways you can do this. One is a plugin. There are tons of plugins out there like https://wordpress.org/plugins/updraftplus/ or https://wordpress.org/plugins/backwpup/ . These plugin can create scheduled backups add you can create and restore backups at any time. The only thing is that they have limited features.

You could go for VaultPress (https://vaultpress.com/) or BackupBuddy (https://ithemes.com/purchase/backupbuddy/). These are easier to use and secure although they come at a price because they are a premium plugin.

Backup in hosting dashboard. You can do this on most hosting companies. They allow you to backup your website although there are next to no settings or extra features.

I would suggest reading through some of these resources if you are still confused.
https://codex.wordpress.org/Child_Themes
https://codex.wordpress.org/WordPress_Backups
https://codex.wordpress.org/Backing_Up_Your_Database
http://www.wpbeginner.com/wp-themes/how-to-create-a-wordpress-child-theme-video/