The menu fonts should change based on your GTK font, but the font used in web pages is a combination of firefox settings and fontconfig.
To change the setting in Firefox for example, go to Edit > Preferences > 'Content' tab > and click the 'Advanced' button next to 'Default font'.
Here you can change the setting for various languages.
If you want to see which font actually gets used for a particular setting in Firefox you can check with the command line: fc-match -s FONT_NAME
This shows the fallback order too, so if characters are missing from the first one in the list then the next font in the list will be used, and so on.
If you want to change this order you can look up how to modify the fontconfig for your Linux distribution, where there are options for system-wide configuration as well as per-user configurations via a file like ~/.config/fontconfig/fonts.conf
If you want to verify your settings in Firefox you can visit a page, right click some text and select "Inspect Element", click the "Inspector" tab, and then click the "Fonts" tab in the smaller window on the right. This will show all fonts being used by firefox for a particular element.