Why does a GFCI trip when anything is plugged into the other outlets?
A GFCI trips when the current coming in through the hot and out the neutral are not equal.  The fact that your GFCI trips whenever something is plugged into a LOAD-side outlet, but not when plugged into the GFCI itself, suggests to me that there's a neutral-to-ground-fault (neutral wire touching ground) somewhere on the LOAD side.



This is the way I would diagnose your problem:


Hit the "Reset" button on the GFCI so the LINE and LOAD are disconnected.
Turn off the electricity to that circuit.
Remove the outlet approximately halfway down the chain on the LOAD side.
Test for continuity (eg. using the OHM setting on a multimeter) between the ground and the neutral screws.  Under normal circumstances, ground and neutral are connected at the main panel, but since the GFCI (hopefully) disconnected the neutral, a correctly-wired circuit should find no continuity.  If my guess is correct, you will find continuity, confirming there is a neutral-to-ground fault.
Look for any loose wires in the box.  If you find any, re-splice the connections.
Unsplice the neutral and ground wires in the box. Check for continuity between the neutral and ground screws again.  If there is continuity, the outlet is defective.
Check for continuity between the incoming neutral/ground wires.  Repeat for the outgoing neutral/ground wires, if any.  If there is continuity, that tells you which half of the circuit the problem is on, so you can repeat this process from step 3 to narrow down the location of the fault.
If the issue turns out to be in the wall between two outlets, I would just run a new wire between those outlets.


The above is not a one-hour job; it will probably take you the better part of a day.  It will take longer if you don't already know how the connections between the outlets are laid out.

If you're not comfortable with any of this, hire a licensed electrician.