For more detailed information see the X11 documentation.
An easy solution
Probably the most common application that is being used that uses X11 PRIMARY by default is xterm.
This can be configured to copy/paste using the CLIPBOARD instead with by adding this to your $HOME/.Xresources file:
This will be automatically loaded next time you log in, but the changes can be tested and loaded into your current X session using xrdb in merge mode:
xrdb -merge $HOME/.Xresources
Getting used to the change
After making this change note that selected text in GUIs which used to select into PRIMARY may now need a Ctrl-C to copy the text into CLIPBOARD (but once this is done middle-clicking into a terminal will now work).